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\begin{document}
\title[General Functional Inequalities]{Testing for a General Class of
Functional Inequalities}
\thanks{We would like to thank Emmanuel Guerre and participants at numerous
seminars and conferences for their helpful comments. We also thank Kyeongbae
Kim, Koohyun Kwon and Jaewon Lee for capable research assistance. Lee's work
was supported by the European Research Council (ERC-2014-CoG-646917-ROMIA).
Song acknowledges the financial support of Social Sciences and Humanities
Research Council of Canada. Whang's work was supported by the National
Research Foundation of Korea Grant funded by the Korean Government
(NRF-2011-342-B00004) and the SNU Creative Leading Researcher Grant.}
\date{\today}
\author[Lee]{Sokbae Lee$^{1,2}$}
\address{$^1$Department of Economics, Columbia University, 1022 International Affairs Building
420 West 118th Street, New York, NY 10027, USA.}
\address{$^2$Centre for Microdata Methods and Practice, Institute for Fiscal
Studies, 7 Ridgmount Street, London, WC1E 7AE, UK.}
\email{sl3841@columbia.edu}
\author[Song]{Kyungchul Song$^3$}
\address{$^3$Vancouver School of Economics, University of British Columbia,
997 - 1873 East Mall, Vancouver, BC, V6T 1Z1, Canada}
\email{kysong@mail.ubc.ca}
\author[Whang]{Yoon-Jae Whang$^4$}
\address{$^4$Department of Economics, Seoul National University, 1
Gwanak-ro, Gwanak-gu, Seoul, 151-742, Republic of Korea.}
\email{whang@snu.ac.kr}


%\maketitle

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\section*{\protect\large Online Appendices to ``Testing for a General Class
of Functional Inequalities''}

Appendix \ref{appendix-proof-A} gives the proofs of Theorems \ref{Thm1}-\ref%
{Thm5}, and Appendices \ref{appendix:C} and \ref{appendix:D} offer auxiliary results
for the proofs of Theorems \ref{Thm1}-\ref{Thm5}.
Appendix \ref{sec:appendix-D} contains the proof of Theorem AUC\ref{thm:AUC1}.
In Appendix \ref{sec:app:literature}, we discuss 
potential areas of applications of our test.

%\clearpage

\appendix

\section{Proofs of Theorems \protect\ref{Thm1}-\protect\ref{Thm5}}

\label{appendix-proof-A}

The roadmap of Appendix A is as follows. Appendix A begins with the proofs
of Lemma 1 (the representation of $\hat{\theta}$) and Lemma 2 (the uniform
convergence of $\hat{v}_{\tau ,j}(x)$). Then we establish auxiliary results,
Lemmas A1-A4, to prepare for the proofs of Theorems 1-3. The brief
descriptions of these auxiliary results are given below.

Lemma A1 establishes asymptotic representation of the location normalizers
for the test statistic both in the population and in the bootstrap
distribution. The crucial implication is that the difference between the
population version and the bootstrap version is of order $o_{P}(h^{d/2})$, $%
\mathcal{P}$-uniformly. The result is in fact an immediate consequence of
Lemma C12 in Appendix C.

Lemma A2 establishes uniform asymptotic normality of the representation of $%
\hat{\theta}$ and its bootstrap version. The asymptotic normality results
use the method of Poissonization as in \citeasnoun{GMZ} and \citeasnoun{LSW}%
. However, in contrast to the preceding research, the results established
here are much more general, and hold uniformly over a wide class of
probabilities. The lemma relies on Lemmas B7-B9 in Appendix B and their
bootstrap versions in Lemmas C7-C9 in Appendix C. These results are employed
to obtain the uniform asymptotic normality of the representation of $\hat{%
\theta}$ in Lemma A2.

Lemma A3 establishes that the estimated contact sets $\hat{B}_{A}(\hat{c}%
_{n})$ are covered by its enlarged population version, and covers its shrunk
population version with probability approaching one uniformly over $P\in 
\mathcal{P}$. In fact, this is an immediate consequence of the uniform
convergence results for $\hat{v}_{\tau ,j}(x)$ and $\hat{\sigma}_{\tau
,j}(x) $ in Assumptions 3 and 5. Lemma A3 is used later, when we replace the
estimated contact sets by their appropriate population versions, eliminating
the nuisance to deal with the estimation errors in $\hat{B}_{A}(\hat{c}_{n})$%
.

Lemma A4 presents the approximation result of the critical values for the
original and bootstrap test statistics in Lemma A2, by critical values from
the standard normal distribution uniformly over $P\in \mathcal{P}.$ Although
we do not propose using the normal critical values, the result is used as an
intermediate step for justifying the use of the bootstrap method in this
paper. Obviously, Lemma A4 follows as a consequence of Lemma A2.

Equipped with Lemmas A1-A4, we proceed to prove Theorem 1. For this, we
first use the representation result of Lemma 1 for $\hat{\theta}$. In doing
so, we use $B_{A}(c_{n,L},c_{n,U})$ as a population version of $\hat{B}_{A}(%
\hat{c}_{n})$. This is because%
\begin{equation*}
B_{A}(c_{n,L},c_{n,U})\subset \hat{B}_{A}(\hat{c}_{n})
\end{equation*}%
with probability approaching one by Lemma A3, and thus, makes the bootstrap
test statistic $\hat{\theta}^{\ast }$ dominate the one that involves $%
B_{A}(c_{n,L},c_{n,U})$ in place of $\hat{B}_{A}(\hat{c}_{n})$. The
distribution of the latter bootstrap version with $B_{A}(c_{n,L},c_{n,U})$
is asymptotically equivalent to the representation of $\hat{\theta}$ with $%
B_{A}(c_{n,L},c_{n,U})$ after location-scale normalization, as long as the
limiting distribution is nondegenerate. When the limiting distribution is
degenerate, we use the second component $h^{d/2}\eta +\hat{a}^{\ast }$ in
the definition of $c_{\alpha ,\eta }^{\ast }$ to ensure the asymptotic
validity of the bootstrap procedure. For both cases of degenerate and
nondegenerate limiting distributions, Lemma A1 which enables one to replace $%
\hat{a}^{\ast }$ by an appropriate population version is crucial.

The proof of Theorem 2 that shows the asymptotic exactness of the bootstrap
test modifies the proof of Theorem 1 substantially. Instead of using the
representation result of Lemma 1 for $\hat{\theta}$ with $%
B_{n,A}(c_{n,L},c_{n,U})$, we now use the same version but with $%
B_{n,A}(c_{n,U},c_{n,L})$. This is because for asymptotic exactness, we need
to approximate the original and bootstrap quantities by versions using $%
B_{n,A}(q_{n})$ for small $q_{n}$, and to do this, we need to control the
remainder term in the bootstrap statistic with the integral domain $\hat{B}%
_{A}(\hat{c}_{n})\backslash B_{n,A}(q_{n})$. By our choice of $%
B_{n,A}(c_{n,U},c_{n,L})$ and by the fact that we have 
\begin{equation*}
\hat{B}_{A}(\hat{c}_{n})\subset B_{n,A}(c_{n,U},c_{n,L}),
\end{equation*}%
with probability approaching one by Lemma A3, we can bound the remainder
term with a version with the integral domain $B_{n,A}(c_{n,U},c_{n,L})%
\backslash B_{n,A}(q_{n})$. Thus this remainder term vanishes by the
condition for $\lambda _{n}$ and $q_{n}$ in the definition of $\mathcal{P}%
_{n}(\lambda _{n},q_{n})$.

The rest of the proofs are devoted to proving the power properties of the
bootstrap procedure. Theorem 3 establishes consistency of the bootstrap
test. Theorems 4 and 5 establish local power functions under Pitman local
drifts. The proofs of Theorems 4-5 are similar to the proof of Theorem 2, as
we need to establish the asymptotically exact form of the rejection
probability for the bootstrap test statistic. Nevertheless, we need to
employ some delicate arguments to deal with the Pitman local alternatives,
and need to expand the rejection probability to obtain the final results.
For this, we first establish Lemmas A5-A7. Essentially, Lemma A5 is a
version of the representation result of Lemma 1 under local alternatives.
Lemma A6 and Lemma A7 parallel Lemma A1 and Lemma 2 under local
alternatives.\bigskip

Let us begin by proving Lemma 1. First, recall the following definitions%
\begin{equation}
\mathbf{\hat{s}}_{\tau }(x)\equiv \left[ \frac{r_{n,j}\{\hat{v}_{\tau
,j}(x)-v_{n,\tau ,j}(x)\}}{\hat{\sigma}_{\tau ,j}(x)}\right] _{j\in \mathbb{N%
}_{J}}\text{ and\ }\mathbf{\hat{s}}_{\tau }^{\ast }(x)\equiv \left[ \frac{%
r_{n,j}\{\hat{v}_{\tau ,j}^{\ast }(x)-\hat{v}_{,\tau ,j}(x)\}}{\hat{\sigma}%
_{\tau ,j}^{\ast }(x)}\right] _{j\in \mathbb{N}_{J}}.  \label{s and s_star}
\end{equation}%
Also, define 
\begin{equation}
\mathbf{\hat{u}}_{\tau }(x)\equiv \left[ \frac{r_{n,j}\hat{v}_{\tau ,j}(x)}{%
\hat{\sigma}_{\tau ,j}(x)}\right] _{j\in \mathbb{N}_{J}}\text{ and\ }\mathbf{%
u}_{\tau }(x;\hat{\sigma})\equiv \left[ \frac{r_{n,j}v_{n,\tau ,j}(x)}{\hat{%
\sigma}_{\tau ,j}(x)}\right] _{j\in \mathbb{N}_{J}}.  \label{v}
\end{equation}

\begin{proof}[Proof of Lemma 1]
It suffices to show the following two statements:

\textbf{Step 1:} As\textit{\ }$n\rightarrow \infty ,$%
\begin{equation*}
\inf_{P\in \mathcal{P}_{0}}P\left\{ \int_{\mathcal{S}\backslash
B_{n}(c_{n,1},c_{n,2})}\Lambda _{p}\left( \mathbf{\hat{u}}_{\tau }(x)\right)
dQ(x,\tau )=0\right\} \rightarrow 1\text{\textit{,}}
\end{equation*}%
where we recall $B_{n}(c_{n,1},c_{n,2})\equiv \cup _{A\in \mathcal{N}%
_{J}}B_{n,A}(c_{n,1},c_{n,2})$.

\textbf{Step 2:} For each $A\in \mathcal{N}_{J}$, as $n\rightarrow \infty ,$%
\begin{equation*}
\inf_{P\in \mathcal{P}_{0}}P\left\{ \int_{B_{n,A}(c_{n,1},c_{n,2})}\left\{
\Lambda _{p}\left( \mathbf{\hat{u}}_{\tau }(x)\right) -\Lambda _{A,p}\left( 
\mathbf{\hat{u}}_{\tau }(x)\right) \right\} dQ(x,\tau )=0\right\}
\rightarrow 1\text{.}
\end{equation*}%
First, we prove Step 1. We write the integral in the probability as%
\begin{equation}
\int_{\mathcal{S}\backslash B_{n}(c_{n,1},c_{n,2})}\Lambda _{p}\left( 
\mathbf{\hat{s}}_{\tau }(x)+\mathbf{u}_{\tau }(x;\hat{\sigma})\right)
dQ(x,\tau ).  \label{eq35}
\end{equation}
Let 
\begin{equation*}
A_{n}(x,\tau )\equiv \left\{ j\in \mathbb{N}_{J}:\frac{r_{n,j}v_{n,\tau
,j}(x)}{\sigma _{n,\tau ,j}(x)}\geq - (c_{n,1}\wedge c_{n,2}) \right\} .
\end{equation*}
We first show that when $(x,\tau )\in \mathcal{S}\backslash
B_{n}(c_{n,1},c_{n,2})$, we have $A_{n}(x,\tau )=\varnothing $ under the
null hypothesis. Suppose that $(x,\tau )\in \mathcal{S}\backslash
B_{n}(c_{n,1},c_{n,2})$ but to the contrary, $A_{n}(x,\tau )$ is nonempty.
By the definition of $A_{n}(x,\tau )$, we have $(x,\tau )\in
B_{n,A_{n}(x,\tau )}(c_{n,1},c_{n,2})$. However, since%
\begin{equation*}
\mathcal{S}\backslash B_{n}(c_{n,1},c_{n,2})=\mathcal{S}\cap \left( \cap
_{A\in \mathcal{N}_{J}}B_{n,A}^{c}(c_{n,1},c_{n,2})\right) \subset
B_{n,A_{n}(x,\tau )}^{c}(c_{n,1},c_{n,2}),
\end{equation*}%
this contradicts the fact that $(x,\tau )\in \mathcal{S}\backslash
B_{n}(c_{n,1},c_{n,2})$. Hence whenever $(x,\tau )\in \mathcal{S}\backslash
B_{n}(c_{n,1},c_{n,2})$, we have $A_{n}(x,\tau )=\varnothing $.

Note that%
\begin{equation*}
\frac{v_{n,\tau ,j}(x)}{\hat{\sigma}_{\tau ,j}(x)}=\frac{v_{n,\tau ,j}(x)}{%
\sigma _{n,\tau ,j}(x)}\left\{ 1+\frac{\sigma _{n,\tau ,j}(x)-\hat{\sigma}%
_{\tau ,j}(x)}{\hat{\sigma}_{\tau ,j}(x)}\right\} =\frac{v_{n,\tau ,j}(x)}{%
\sigma _{n,\tau ,j}(x)}\left\{ 1+o_{P}(1)\right\} ,
\end{equation*}%
where $o_{P}(1)$ is uniform over $(x,\tau )\in \mathcal{S}$ and over $P\in 
\mathcal{P}$ by Assumption A\ref{assumption-A5}. Fix a small $\varepsilon >0$%
. We have for all $j\in \mathbb{N}_{J},$%
\begin{eqnarray*}
&&\inf_{P\in \mathcal{P}_{0}}P\left\{ \frac{r_{n,j}v_{n,\tau ,j}(x)}{\hat{%
\sigma}_{\tau ,j}(x)}<-\frac{c_{n,1}\wedge c_{n,2}}{1+\varepsilon }\text{
for all }(x,\tau )\in \mathcal{S}\backslash B_{n}(c_{n,1},c_{n,2})\right\} \\
&\geq &\inf_{P\in \mathcal{P}_{0}}P\left\{ \frac{r_{n,j}v_{n,\tau ,j}(x)}{%
\sigma _{n,\tau ,j}(x)}<-\frac{c_{n,1}\wedge c_{n,2}}{(1+\varepsilon
)\left\{ 1+o_{P}(1)\right\} }\text{ for all }(x,\tau )\in \mathcal{S}%
\backslash B_{n}(c_{n,1},c_{n,2})\right\} \rightarrow 1,
\end{eqnarray*}%
as $n\rightarrow \infty $, where the last convergence follows because $%
A_{n}(x,\tau )=\varnothing $ for all $(x,\tau )\in \mathcal{S}\backslash
B_{n}(c_{n,1},c_{n,2})$. Therefore, with probability approaching one, the
term in (\ref{eq35}) is bounded by%
\begin{equation}
\int_{\mathcal{S}\backslash B_{n}(c_{n,1},c_{n,2})}\Lambda _{p}\left( 
\mathbf{\hat{s}}_{\tau }(x)-\left( \frac{c_{n,1}\wedge c_{n,2}}{%
1+\varepsilon }\right) \mathbf{1}_{J}\right) dQ(x,\tau ),  \label{inet}
\end{equation}%
where $\mathbf{1}_{J}$ is a $J$-dimensional vector of ones. Using the
definition of $\Lambda _{p}(\mathbf{v})$, bound the above integral by%
\begin{equation}
J^{p/2}\left( \sum_{j=1}^{J}\left[ r_{n,j}\sup_{(x,\tau )\in \mathcal{S}%
}\left\vert \frac{\hat{v}_{\tau ,j}(x)-v_{n,\tau ,j}(x)}{\hat{\sigma}_{\tau
,j}(x)}\right\vert -\frac{c_{n,1}\wedge c_{n,2}}{1+\varepsilon }\right]
_{+}^{2}\right) ^{p/2}.  \label{dec45}
\end{equation}%
Note that by Assumption A\ref{assumption-A3}, 
\begin{equation*}
r_{n,j}\sup_{(x,\tau )\in \mathcal{S}}\left\vert \frac{\hat{v}_{\tau
,j}(x)-v_{n,\tau ,j}(x)}{\hat{\sigma}_{\tau ,j}(x)}\right\vert =O_{P}\left( 
\sqrt{\log n}\right) .
\end{equation*}%
Fix any arbitrarily large $M>0$ and denote by $E_{n}$ the event that%
\begin{equation*}
r_{n,j}\sup_{(x,\tau )\in \mathcal{S}}\left\vert \frac{\hat{v}_{\tau
,j}(x)-v_{n,\tau ,j}(x)}{\hat{\sigma}_{\tau ,j}(x)}\right\vert \leq M\sqrt{%
\log n}.
\end{equation*}%
The term (\ref{dec45}), when restricted to this event $E_{n},$ is bounded by%
\begin{equation*}
J^{p/2}\left( \sum_{j=1}^{J}\left[ M\sqrt{\log n}-\frac{c_{n,1}\wedge c_{n,2}%
}{1+\varepsilon }\right] _{+}^{2}\right) ^{p/2}
\end{equation*}%
which becomes zero from some large $n$ on, given that $(c_{n,1}\wedge
c_{n,2})/\sqrt{\log n}\rightarrow \infty $. Since sup$_{P\in \mathcal{P}%
_{0}}PE_{n}^{c}\rightarrow 0$ as $n\rightarrow \infty $ and then $%
M\rightarrow \infty $ by Assumption A3, we obtain the desired result of Step
1.

As for Step 2, we have for any small $\varepsilon >0$, and for all $j\in 
\mathbb{N}_{J}\backslash A$,%
\begin{eqnarray}
&&P\left\{ \frac{r_{n,j}v_{n,\tau ,j}(x)}{\hat{\sigma}_{\tau ,j}(x)}<-\frac{%
c_{n,1}\wedge c_{n,2}}{1+\varepsilon }\text{ for all }(x,\tau )\in
B_{n,A}(c_{n,1},c_{n,2})\right\}  \label{ineq54} \\
&\geq &P\left\{ \frac{r_{n,j}v_{n,\tau ,j}(x)}{\sigma _{n,\tau ,j}(x)}<-%
\frac{c_{n,1}\wedge c_{n,2}}{(1+\varepsilon )\left\{ 1+o_{P}(1)\right\} }%
\text{ for all }(x,\tau )\in B_{n,A}(c_{n,1},c_{n,2})\right\} \rightarrow 1,
\notag
\end{eqnarray}%
similarly as before. Let $\mathbf{\bar{s}}_{\tau ,A}(x)$ be a $J$%
-dimensional vector whose $j$-th entry is $r_{n,j}\hat{v}_{n,\tau ,j}(x)/%
\hat{\sigma}_{\tau ,j}(x)$ if $j\in A$, and $r_{n,j}\{\hat{v}_{n,\tau
,j}(x)-v_{n,\tau ,j}(x)\}/\hat{\sigma}_{\tau ,j}(x)$ if $j\in \mathbb{N}%
_{J}\backslash A$. Since by Assumption A5, we have 
\begin{equation*}
\inf_{P\in \mathcal{P}_{0}}P\left\{ \mathbf{u}_{\tau }(x;\hat{\sigma})\leq 0%
\text{ for all }(x,\tau )\in \mathcal{S}\right\} \rightarrow 1,
\end{equation*}%
as $n\rightarrow \infty $, using either definition of $\Lambda _{p}(\mathbf{v%
})$ in \eqref{lambda_p}, we find that with probability approaching one (uniformly over $P \in \mathcal{P}_0$), 
\begin{eqnarray}
&&\int_{B_{n,A}(c_{n,1},c_{n,2})}\Lambda _{A,p}\left( \mathbf{\hat{u}}_{\tau
}(x)\right) dQ(x,\tau )  \label{ineqs2} \\
&\leq &\int_{B_{n,A}(c_{n,1},c_{n,2})}\Lambda _{p}\left( \mathbf{\hat{u}}%
_{\tau }(x)\right) dQ(x,\tau )  \notag \\
&\leq &\int_{B_{n,A}(c_{n,1},c_{n,2})}\Lambda _{p}\left( \mathbf{\bar{s}}%
_{\tau ,A}(x)-\frac{c_{n,1}\wedge c_{n,2}}{1+\varepsilon }\mathbf{1}%
_{-A}\right) dQ(x,\tau ),  \notag
\end{eqnarray}%
where $\mathbf{1}_{-A}$ is the $J$-dimensional vector whose $j$-th entry is
zero if $j\in A$ and one if $j\in \mathbb{N}_{J}\backslash A$, and the last
inequality holds with probability approaching one by (\ref{ineq54}). Note
that by Assumption A\ref{assumption-A3} and by the assumption that $\sqrt{%
\log n}\{c_{n,1}^{-1}+c_{n,2}^{-1}\}\rightarrow \infty $, we deduce that for
any $j\in \mathbb{N}_{J},$%
\begin{equation*}
\inf_{P\in \mathcal{P}_{0}}P\left\{ r_{n,j}\sup_{(x,\tau )\in \mathcal{S}%
}\left\vert \frac{\hat{v}_{\tau ,j}(x)-v_{n,\tau ,j}(x)}{\hat{\sigma}_{\tau
,j}(x)}\right\vert \leq \frac{c_{n,1}\wedge c_{n,2}}{1+\varepsilon }\right\}
\rightarrow 1,
\end{equation*}%
as $n\rightarrow \infty $. Hence, as $n\rightarrow \infty ,$%
\begin{equation*}
\inf_{P\in \mathcal{P}_{0}}P\left\{ 
\begin{array}{c}
\int_{B_{n,A}(c_{n,1},c_{n,2})}\Lambda _{p}\left( \mathbf{\bar{s}}_{\tau
,A}(x)-((c_{n,1}\wedge c_{n,2})/(1+\varepsilon ))\mathbf{1}_{-A}\right)
dQ(x,\tau ) \\ 
=\int_{B_{n,A}(c_{n,1},c_{n,2})}\Lambda _{A,p}\left( \mathbf{\bar{s}}_{\tau
,A}(x)\right) dQ(x,\tau )%
\end{array}%
\right\} \rightarrow 1.
\end{equation*}%
Since%
\begin{equation*}
\int_{B_{n,A}(c_{n,1},c_{n,2})}\Lambda _{A,p}\left( \mathbf{\bar{s}}_{\tau
,A}(x)\right) dQ(x,\tau )=\int_{B_{n,A}(c_{n,1},c_{n,2})}\Lambda
_{A,p}\left( \mathbf{\hat{u}}_{\tau }(x)\right) dQ(x,\tau ),
\end{equation*}%
we obtain the desired result from (\ref{ineqs2}).
\end{proof}

Now let us turn to the proof of Lemma 2 in Section 4.4.

\begin{proof}[Proof of Lemma 2]
(i) Recall the definition $b_{n,ij}(x,\tau )\equiv \beta _{n,x,\tau
,j}\left( Y_{ij},(X_{i}-x)/h\right) )$. Take $M_{n,j}\equiv \sqrt{nh^{d}}/%
\sqrt{\log n},$ and let%
\begin{equation*}
b_{n,ij}^{a}(x,\tau )\equiv b_{n,ij}(x,\tau )1_{n,ij}\text{ and\ }%
b_{n,ij}^{b}(x,\tau )\equiv b_{n,ij}(x,\tau )\left( 1-1_{n,ij}\right) ,
\end{equation*}%
where $1_{n,ij}\equiv 1\{$sup$_{(x,\tau )\in \mathcal{S}}|b_{n,ij}(x,\tau
)|\leq M_{n,j}/2\}.$ First, note that by Assumption A1,%
\begin{eqnarray}
&&r_{n,j}\sqrt{h^{d}}\sup_{(x,\tau )\in \mathcal{S}}\left\vert \frac{\hat{v}%
_{\tau ,j}(x)-v_{n,\tau ,j}(x)}{\hat{\sigma}_{\tau ,j}(x)}\right\vert
\label{decomp2} \\
&\leq &\sup_{(x,\tau )\in \mathcal{S}}\left\vert \frac{1}{\sqrt{n}}%
\sum_{i=1}^{n}\left( b_{n,ij}^{a}(x,\tau )-\mathbf{E}\left[
b_{n,ij}^{a}(x,\tau )\right] \right) \right\vert  \notag \\
&&+\sup_{(x,\tau )\in \mathcal{S}}\left\vert \frac{1}{\sqrt{n}}%
\sum_{i=1}^{n}\left( b_{n,ij}^{b}(x,\tau )-\mathbf{E}\left[
b_{n,ij}^{b}(x,\tau )\right] \right) \right\vert +o_{P}(1),\ \mathcal{P}%
\text{-uniformly.}
\end{eqnarray}%
We now prove part (i) by proving the following two steps.\newline
\textbf{Step 1:}%
\begin{equation*}
\sup_{(x,\tau )\in \mathcal{S}}\left\vert \frac{1}{\sqrt{nh^{d}}}%
\sum_{i=1}^{n}\left( b_{n,ij}^{b}(x,\tau )-\mathbf{E}\left[
b_{n,ij}^{b}(x,\tau )\right] \right) \right\vert =o_{P}(\sqrt{\log n}),\ 
\mathcal{P}\text{-uniformly.}
\end{equation*}%
\newline
\textbf{Step 2:}%
\begin{equation*}
\sup_{(x,\tau )\in \mathcal{S}}\left\vert \frac{1}{\sqrt{nh^{d}}}%
\sum_{i=1}^{n}\left( b_{n,ij}^{a}(x,\tau )-\mathbf{E}\left[
b_{n,ij}^{a}(x,\tau )\right] \right) \right\vert =O_{P}(\sqrt{\log n}),\ 
\mathcal{P}\text{-uniformly.}
\end{equation*}%
\newline
Step 1 is carried out by some elementary moment calculations, whereas Step 2
is proved using a maximal inequality of \citeasnoun[Theorem
6.8]{Massart:07}.

\textbf{Proof of Step 1:} It is not hard to see that%
\begin{eqnarray*}
&&\mathbf{E}\left[ \sup_{(x,\tau )\in \mathcal{S}}\left\vert \frac{1}{\sqrt{n%
}}\sum_{i=1}^{n}\left( b_{n,ij}^{b}(x,\tau )-\mathbf{E}\left[
b_{n,ij}^{b}(x,\tau )\right] \right) \right\vert \right] \\
&\leq &2\sqrt{n}\mathbf{E}\left[ \sup_{(x,\tau )\in \mathcal{S}}\left\vert
b_{n,ij}(x,\tau )\right\vert \left( 1-1_{n,ij}\right) \right] \\
&\leq &C\sqrt{n}\left( \frac{M_{n,j}}{2}\right) ^{-3}\mathbf{E}\left[
\sup_{(x,\tau )\in \mathcal{S}}\left\vert b_{n,ij}(x,\tau )\right\vert ^{4}%
\right] \leq C_{1}\sqrt{n}\left( \frac{M_{n,j}}{2}\right) ^{-3},
\end{eqnarray*}%
for some $C_{1}>0$, $C>0$. The last bound follows by the uniform fourth
moment bound for $b_{n,ij}(x,\tau )$ assumed in Lemma 2. Note that 
\begin{equation*}
\sqrt{n}\left( M_{n,j}\right) ^{-3}=n^{-1}h^{-3d/2}\left( \log n\right)
^{3/2}=o\left( \sqrt{\log n}h^{d/2}\right) ,
\end{equation*}%
by the condition that $n^{-1/2}h^{-d-\nu }\rightarrow 0$ for some small $\nu
>0$.

\textbf{Proof of Step 2: }For each $j\in \mathbb{N}_{J}$, let $\mathcal{F}%
_{n,j}\equiv \{\beta _{n,x,\tau ,j}^{a}(\cdot ,(\cdot -x)/h)/M_{n,j}:(x,\tau
)\in \mathcal{S}\}$, where $\beta _{n,x,\tau
,j}^{a}(Y_{ij},(X_{i}-x)/h)\equiv b_{n,ij}^{a}(x,\tau )$. Note that the
indicator function $1_{n,ij}$ in the definition of $\beta _{n,x,\tau ,j}^{a}$
does not depend on $(x,\tau )$ of $\beta _{n,x,\tau ,j}^{a}$. Using (\ref%
{Lp-Cont}) in Lemma 2 and following (part of) the arguments in the proof of
Theorem 3 of \citeasnoun{Chen/Linton/VanKeilegom:03}, we find that there
exist $C_{1}>0$ and $C_{2,j}>0$ such that for all $\varepsilon >0,$%
\begin{equation*}
N_{[]}\left( \varepsilon ,\mathcal{F}_{n,j},L_{2}(P)\right) \leq N\left(
\left( \frac{\varepsilon M_{n,j}}{\delta _{n,j}}\right) ^{2/\gamma _{j}},%
\mathcal{X}\times \mathcal{T},||\cdot ||\right) \leq C_{1}\left( \frac{%
\varepsilon M_{n,j}}{\delta _{n,j}}\wedge 1\right) ^{-C_{2,j}},
\end{equation*}%
where $N_{[]}\left( \varepsilon ,\mathcal{F}_{n,j},L_{2}(P)\right) $ denotes
the $\varepsilon $-bracketing number of the class $\mathcal{F}_{n,j}$ with
respect to the $L_{2}(P)$-norm and $N\left( \varepsilon ,\mathcal{X}\times 
\mathcal{T},||\cdot ||\right) $ denotes the $\varepsilon $-covering number
of the space $\mathcal{X}\times \mathcal{T}$ with respect to the Euclidean
norm $||\cdot ||$. The last inequality follows by the assumption that $%
\mathcal{X}$ and $\mathcal{T}$ are compact subsets of a Euclidean space. The
class $\mathcal{F}_{n,j}$ is uniformly bounded by $1/2$. \newline
Let $\{[\beta _{n,x_{k},\tau _{k},j}^{a}(\cdot ,(\cdot
-x_{k})/h)/M_{n,j}-\Delta _{k}(\cdot ,\cdot )/M_{n,j},\beta _{n,x_{k},\tau
_{k},j}^{a}(\cdot ,(\cdot -x_{k})/h)/M_{n,j}+\Delta _{k}(\cdot ,\cdot
)/M_{n,j}]:k=1,...,N_{n,j}\}$ constitutes $\varepsilon $%
-brackets, where $\Delta _{k}(Y_{ij},X_{i})\equiv \sup |\beta _{n,x,\tau
,j}^{a}(Y_{ij},(X_{i}-x)/h)-\beta _{n,x_{k},\tau
_{k},j}^{a}(Y_{ij},(X_{i}-x_{k})/h)|$ and the supremum is over $(x,\tau )\in 
\mathcal{S}$ such that 
\begin{equation*}
\sqrt{||x-x_{k}||^{2}+||\tau -\tau _{k}||^{2}}\leq C_{1}(\varepsilon
M_{n,j}/\delta _{n,j})^{2/\gamma _{j}}.
\end{equation*}%
By the previous covering number bound, we can take $N_{n,j}\leq C_{1}\left(
(\varepsilon M_{n,j}/\delta _{n,j})\wedge 1\right) ^{-C_{2,j}},$ and 
\begin{equation*}
\mathbf{E}\Delta _{k}^{2}(Y_{ij},X_{i})M_{n,j}^{-2}<\varepsilon ^{2}.
\end{equation*}%

Note that for any $k\geq 2,$%
\begin{equation*}
\mathbf{E}\left[ |b_{n,ij}^{a}(x,\tau )/M_{n,j}|^{k}\right] \leq \mathbf{E}%
\left[ b_{n,ij}^{2}(x,\tau )\right] /M_{n,j}^{2}\leq
CM_{n,j}^{-2}h^{d}=C(\log n)/n\text{,}
\end{equation*}%
by the fact that $|b_{n,ij}^{a}(x,\tau )/M_{n,j}|\leq 1/2$. Furthermore,%
\begin{equation*}
\mathbf{E}\left[ |\Delta _{k}(Y_{ij},X_{i})/M_{n,j}|^{k}\right] \leq \mathbf{%
E}\left[ \Delta _{k}^{2}(Y_{ij},X_{i})/M_{n,j}^{2}\right] \leq \varepsilon
^{2},
\end{equation*}%
where the first inequality follows because $|\Delta
_{k}(Y_{ij},X_{i})/M_{n,j}|\leq 1$. Therefore, by Theorem 6.8 of %
\citeasnoun{Massart:07}, we have (from sufficiently large $n$ on)%
\begin{eqnarray}
&&\mathbf{E}\left[ \sup_{(x,\tau )\in \mathcal{S}}\left\vert \frac{1}{M_{n,j}%
\sqrt{n}}\sum_{i=1}^{n}\left( b_{n,ij}^{a}(x,\tau )-\mathbf{E}\left[
b_{n,ij}^{a}(x,\tau )\right] \right) \right\vert \right]  \label{bd5} \\
&\leq &C_{1}\int_{0}^{\frac{C_{2}h^{d/2}}{M_{n,j}}}\left\{ \left( -C_{3}\log
\left( \frac{\varepsilon M_{n,j}}{\delta _{n,j}}\wedge 1\right) \right)
\wedge n\right\} ^{1/2}d\varepsilon -\frac{C_{4}}{\sqrt{n}}\log \left( \frac{%
\sqrt{\log n}}{\sqrt{n}}\right) ,  \notag
\end{eqnarray}%
where $C_{1},C_{2},C_{3}$, and $C_{4}$ are positive constants. (The
inequality above follows because$\sqrt{\log n}/\sqrt{n}\rightarrow 0$ as $%
n\rightarrow \infty $.) The leading integral has a domain restricted to $%
[0,\delta _{n,j}/M_{n,j}]$, so that it is equal to%
\begin{eqnarray*}
&&C_{1}\int_{0}^{\frac{C_{2}h^{d/2}}{M_{n,j}}\wedge \frac{\delta _{n,j}}{%
M_{n,j}}}\left\{ \left( -C_{3}\log \left( \frac{\varepsilon M_{n,j}}{\delta
_{n,j}}\right) \right) \wedge n\right\} ^{1/2}d\varepsilon \\
&=&\frac{C_{1}\delta _{n,j}}{M_{n,j}}\int_{0}^{\frac{C_{2}h^{d/2}}{\delta
_{n,j}}\wedge 1}\sqrt{\left( -C_{3}\log \varepsilon \right) \wedge n}%
d\varepsilon \\
&=&O\left( \frac{\delta _{n,j}}{M_{n,j}}\left( \frac{h^{d/2}}{\delta _{n,j}}%
\wedge 1\right) \sqrt{-\log \left( \frac{h^{d/2}}{\delta _{n,j}}\wedge
1\right) }\right) .
\end{eqnarray*}%
After multiplying by $M_{n,j}/h^{d/2}$, the last term is of order%
\begin{equation*}
O\left( \left( 1\wedge \frac{\delta _{n,j}}{h^{d/2}}\right) \sqrt{-\log
\left( \frac{h^{d/2}}{\delta _{n,j}}\wedge 1\right) }\right) =O\left( \sqrt{%
-\log \left( \frac{h^{d/2}}{\delta _{n,j}}\wedge 1\right) }\right) =O(\sqrt{%
\log n}),
\end{equation*}%
because $\delta _{n,j}=n^{s_{1,j}}$ and $h=n^{s_{2}}$ for some $%
s_{1,j},s_{2}\in \mathbf{R}$.

Also, note that after multiplying by $M_{n,j}/h^{d/2}=\sqrt{n}/\sqrt{\log n}$%
, the last term in (\ref{bd5}) (with minus sign) becomes%
\begin{equation*}
-\frac{C_{4}}{\sqrt{\log n}}\log \left( \frac{\sqrt{\log n}}{\sqrt{n}}%
\right) \leq \frac{C_{4}\sqrt{\log n}}{2}-\frac{C_{4}\log \sqrt{\log n}}{%
\sqrt{\log n}}=O\left( \sqrt{\log n}\right) ,
\end{equation*}%
where the inequality follows because $\sqrt{\log n}\geq 1$ for all $n\geq
e\equiv \exp (1)$. Collecting the results for both the terms on the right
hand side of (\ref{bd5}), we obtain the desired result of Step 2.
\smallskip

\noindent (ii) Define $b_{n,ij}^{\ast }(x,\tau )\equiv \beta _{n,x,\tau
,j}(Y_{ij}^{\ast },(X_{i}^{\ast }-x)/h)$. By Assumptions B\ref{assumption-B1}
and B3, it suffices to show that%
\begin{equation*}
\sup_{(x,\tau )\in \mathcal{S}}\left\vert \frac{1}{\sqrt{nh^{d}}}%
\sum_{i=1}^{n}\left( b_{n,ij}^{\ast }(x,\tau )-\mathbf{E}^{\ast }\left[
b_{n,ij}^{\ast }(x,\tau )\right] \right) \right\vert =O_{P^{\ast }}(\sqrt{%
\log n}),\text{ }\mathcal{P}\text{-uniformly.}
\end{equation*}%
Using Le Cam's Poissonization lemma in \citeasnoun{Gine/Zinn:90}
(Proposition 2.2 on p.855) and following the arguments in the proof of
Theorem 2.2 of \citeasnoun{Gine:97}, we deduce that%
\begin{eqnarray*}
&&\mathbf{E}\left[ \mathbf{E}^{\ast }\left( \sup_{(x,\tau )\in \mathcal{S}%
}\left\vert \frac{1}{\sqrt{nh^{d}}}\sum_{i=1}^{n}\left( b_{n,ij}^{\ast
}(x,\tau )-\mathbf{E}^{\ast }\left[ b_{n,ij}^{\ast }(x,\tau )\right] \right)
\right\vert \right) \right] \\
&\leq &\frac{e}{e-1}\mathbf{E}\left[ \sup_{(x,\tau )\in \mathcal{S}%
}\left\vert \frac{1}{\sqrt{nh^{d}}}\sum_{i=1}^{n}\left( N_{i}-1\right)
\left\{ b_{n,ij}(x,\tau )-\frac{1}{n}\sum_{k=1}^{n}b_{n,kj}(x,\tau )\right\}
\right\vert \right] ,
\end{eqnarray*}%
where $N_{i}$'s are i.i.d. Poisson random variables with mean $1$ and
independent of $\{(X_{i},Y_{i})\}_{i=1}^{n}$. The last expectation is
bounded by%
\begin{eqnarray*}
&&\mathbf{E}\left[ \sup_{(x,\tau )\in \mathcal{S}}\left\vert \frac{1}{\sqrt{%
nh^{d}}}\sum_{i=1}^{n}\left\{ \left( N_{i}-1\right) b_{n,ij}(x,\tau )-%
\mathbf{E}\left[ \left( N_{i}-1\right) b_{n,ij}(x,\tau )\right] \right\}
\right\vert \right] \\
&&+\mathbf{E}\left[ \sup_{(x,\tau )\in \mathcal{S}}\left\vert \frac{1}{n}%
\sum_{i=1}^{n}\left( N_{i}-1\right) \right\vert \left\vert \frac{1}{\sqrt{%
nh^{d}}}\sum_{k=1}^{n}\left( b_{n,kj}(x,\tau )-\mathbf{E}\left[
b_{n,kj}(x,\tau )\right] \right) \right\vert \right] .
\end{eqnarray*}%
Using the same arguments as in the proof of (i), we find that the first
expectation is $O\left( \sqrt{\log n}\right) $ uniformly in $P\in \mathcal{P}
$. Using independence, we write the second expectation as 
\begin{equation*}
\mathbf{E}\left[ \left\vert \frac{1}{n}\sum_{i=1}^{n}\left( N_{i}-1\right)
\right\vert \right] \cdot \mathbf{E}\left[ \sup_{(x,\tau )\in \mathcal{S}%
}\left\vert \frac{1}{\sqrt{nh^{d}}}\sum_{k=1}^{n}\left( b_{n,kj}(x,\tau )-%
\mathbf{E}\left[ b_{n,kj}(x,\tau )\right] \right) \right\vert \right]
\end{equation*}%
which, as shown in the proof of part (i), is $O(\sqrt{\log n})$, uniformly
in $P\in \mathcal{P}$.
\end{proof}

For further proofs, we introduce new notation. Define for any positive
sequences $c_{n,1}$ and $c_{n,2}$, and any $\mathbf{v}\in \mathbf{R}^{J}$,%
\begin{equation}
\bar{\Lambda}_{x,\tau }(\mathbf{v})\equiv \sum_{A\in \mathcal{N}_{J}}\Lambda
_{A,p}(\mathbf{v})1\{(x,\tau )\in B_{n,A}(c_{n,1},c_{n,2})\}.  \label{lambe}
\end{equation}%
We let%
\begin{eqnarray}
a_{n}^{R}(c_{n,1},c_{n,2}) &\equiv &\int_{\mathcal{X}\times \mathcal{T}}%
\mathbf{E}\left[ \bar{\Lambda}_{x,\tau }(\sqrt{nh^{d}}\mathbf{z}_{N,\tau
}(x))\right] dQ(x,\tau ),\text{ and}  \label{atilde} \\
a_{n}^{R\ast }(c_{n,1},c_{n,2}) &\equiv &\int_{\mathcal{X}\times \mathcal{T}}%
\mathbf{E}^{\ast }\left[ \bar{\Lambda}_{x,\tau }(\sqrt{nh^{d}}\mathbf{z}%
_{N,\tau }^{\ast }(x))\right] dQ(x,\tau ),  \notag
\end{eqnarray}%
where $\mathbf{z}_{N,\tau }(x)$ and $\mathbf{z}_{N,\tau }^{\ast }(x)$ are
random vectors whose $j$-th entry is respectively given by 
\begin{eqnarray*}
z_{N,\tau ,j}(x) &\equiv &\frac{1}{nh^{d}}\sum_{i=1}^{N}\left( \beta
_{n,x,\tau ,j}\left( Y_{ij},\frac{X_{i}-x}{h}\right) -\mathbf{E}\left[ \beta
_{n,x,\tau ,j}\left( Y_{ij},\frac{X_{i}-x}{h}\right) \right] \right) \text{
and} \\
z_{N,\tau ,j}^{\ast }(x) &\equiv &\frac{1}{nh^{d}}\sum_{i=1}^{N}\left( \beta
_{n,x,\tau ,j}\left( Y_{ij}^{\ast },\frac{X_{i}^{\ast }-x}{h}\right) -%
\mathbf{E}^{\ast }\left[ \beta _{n,x,\tau ,j}\left( Y_{ij}^{\ast },\frac{%
X_{i}^{\ast }-x}{h}\right) \right] \right) ,
\end{eqnarray*}%
and $N$ is a Poisson random variable with mean $n$ and independent of $%
\{Y_{i},X_{i}\}_{i=1}^{\infty }$. We also define%
\begin{equation*}
a_{n}(c_{n,1},c_{n,2})\equiv \int \mathbf{E}\left[ \bar{\Lambda}_{x,\tau }(%
\mathbb{W}_{n,\tau ,\tau }^{(1)}(x,0))\right] dQ(x,\tau ).
\end{equation*}%
(See Section 6.3 for the definition of $\mathbb{W}_{n,\tau ,\tau
}^{(1)}(x,u) $.)

\begin{LemmaA}
Suppose that Assumptions A\ref{assumption-A6}(i) and B\ref{assumption-B4}
hold and let $c_{n,1}$ and $c_{n,2}$ be any nonnegative sequences. Then%
\begin{eqnarray*}
\left\vert a_{n}^{R}(c_{n,1},c_{n,2})-a_{n}(c_{n,1},c_{n,2})\right\vert
&=&o(h^{d/2})\text{, uniformly in }P\in \mathcal{P}\text{, and} \\
\left\vert a_{n}^{R\ast }(c_{n,1},c_{n,2})-a_{n}(c_{n,1},c_{n,2})\right\vert
&=&o_{P}(h^{d/2}),\ \mathcal{P}\text{-uniformly.}
\end{eqnarray*}
\end{LemmaA}

\begin{proof}[Proof of Lemma A1]
The proof is essentially the same as the proof of Lemma C12 in Appendix C.
\end{proof}

For any given nonnegative sequences $c_{n,1},c_{n,2},$ we define%
\begin{equation}
\sigma _{n}^{2}(c_{n,1},c_{n,2})\equiv \int_{\mathcal{T}}\int_{\mathcal{T}%
}\int_{\mathcal{X}}\bar{C}_{\tau _{1},\tau _{2}}(x)dxd\tau _{1}d\tau _{2},
\label{sigmae}
\end{equation}%
where 
\begin{equation*}
\bar{C}_{\tau _{1},\tau _{2}}(x)\equiv \int_{\mathcal{U}}Cov\left( \bar{%
\Lambda}_{n,x,\tau _{1}}(\mathbb{W}_{n,\tau _{1},\tau _{2}}^{(1)}(x,u)),\bar{%
\Lambda}_{n,x,\tau _{2}}(\mathbb{W}_{n,\tau _{1},\tau
_{2}}^{(2)}(x,u))\right) du\text{.}
\end{equation*}

Let%
\begin{equation}
\bar{\theta}_{n}(c_{n,1},c_{n,2})\equiv \int \bar{\Lambda}_{x,\tau }\left( 
\mathbf{\hat{s}}_{\tau }(x)\right) dQ(x,\tau )\text{,}  \label{theta_hate}
\end{equation}%
and%
\begin{equation}
\bar{\theta}_{n}^{\ast }(c_{n,1},c_{n,2})\equiv \int \bar{\Lambda}_{x,\tau
}\left( \mathbf{\hat{s}}_{\tau }^{\ast }(x)\right) dQ(x,\tau ).
\label{theta*e}
\end{equation}

From here on, for any sequence of random quantities $Z_{n}$ and a random
vector $Z$, we write 
\begin{equation*}
Z_{n}\overset{d}{\rightarrow }N(0,1)\text{, }\mathcal{P}_{0}\text{-uniformly,%
}
\end{equation*}%
if for each $t>0$,%
\begin{equation*}
\sup_{P\in \mathcal{P}_{0}}\left\vert P\left\{ Z_{n}\leq t\right\} -\Phi
(t)\right\vert =o(1)\text{.}
\end{equation*}%
And for any sequence of bootstrap quantities $Z_{n}^{\ast }$ and a random
vector $Z$, we write 
\begin{equation*}
Z_{n}^{\ast }\overset{d^{\ast }}{\rightarrow }N(0,1)\text{, }\mathcal{P}_{0}%
\text{-uniformly,}
\end{equation*}%
if for each $t>0$,%
\begin{equation*}
\left\vert P^{\ast }\left\{ Z_{n}^{\ast }\leq t\right\} -\Phi (t)\right\vert
=o_{P^{\ast }}(1)\text{, }\mathcal{P}_{0}\text{-uniformly.}
\end{equation*}

\begin{LemmaA}
(i) \textit{Suppose that Assumptions }A\ref{assumption-A1}-A\ref%
{assumption-A3}, A4(i)\textit{, and }A5-A6\textit{\ are satisfied. Then for
any sequences }$c_{n,1},c_{n,2}>0$\textit{\ such that }$\liminf_{n%
\rightarrow \infty }\inf_{P\in \mathcal{P}_{0}}\sigma
_{n}^{2}(c_{n,1},c_{n,2})>0$ and $\sqrt{\log n}/c_{n,2}\rightarrow 0$, as $%
n\rightarrow \infty ,$%
\begin{equation*}
h^{-d/2}\left( \frac{\bar{\theta}%
_{n}(c_{n,1},c_{n,2})-a_{n}^{R}(c_{n,1},c_{n,2})}{\sigma
_{n}(c_{n,1},c_{n,2})}\right) \overset{d}{\rightarrow }N(0,1),\text{ }%
\mathcal{P}_{0}\text{-\textit{uniformly}}.
\end{equation*}

\noindent (ii) \textit{Suppose that Assumptions A\ref{assumption-A1}-A\ref%
{assumption-A3}, A4(i), A5-A6, B\ref{assumption-B1} and B\ref{assumption-B4}
are satisfied. Then for any sequences }$c_{n,1},c_{n,2}\geq 0$\textit{\ such
that }$\liminf_{n\rightarrow \infty }\inf_{P\in \mathcal{P}_{0}}\sigma
_{n}^{2}(c_{n,1},c_{n,2})>0$ and $\sqrt{\log n}/c_{n,2}\rightarrow 0$, as $%
n\rightarrow \infty ,$%
\begin{equation*}
h^{-d/2}\left( \frac{\bar{\theta}_{n}^{\ast }(c_{n,1},c_{n,2})-a_{n}^{R\ast
}(c_{n,1},c_{n,2})}{\sigma _{n}(c_{n,1},c_{n,2})}\right) \overset{d^{\ast }}{%
\rightarrow }N(0,1)\text{, }\mathcal{P}_{0}\text{-\textit{uniformly}}.
\end{equation*}
\end{LemmaA}

\begin{proof}[Proof of Lemma A2]
(i) By Lemma 1, we have (with probability approaching one)%
\begin{equation*}
\bar{\theta}_{n}(c_{n,1},c_{n,2})=\sum_{A\in \mathcal{N}_{J}}%
\int_{B_{n,A}(c_{n,1},c_{n,2})}\Lambda _{p}(\mathbf{\hat{s}}_{\tau
}(x))dQ(x,\tau )=\sum_{A\in \mathcal{N}_{J}}\int_{B_{n,A}(c_{n,1},c_{n,2})}%
\Lambda _{A,p}(\mathbf{\hat{s}}_{\tau }(x))dQ(x,\tau ).
\end{equation*}%
Note that $a_{n}^{R}(c_{n,1},c_{n,2})=\sum_{A\in \mathcal{N}%
_{J}}a_{n,A}^{R}(c_{n,1},c_{n,2})$, where%
\begin{equation*}
a_{n,A}^{R}(c_{n,1},c_{n,2})\equiv \int_{B_{n,A}(c_{n,1},c_{n,2})}\mathbf{E}%
\left[ \Lambda _{A,p}(\sqrt{nh^{d}}\mathbf{z}_{N,\tau }(x))\right] dQ(x,\tau
).
\end{equation*}%
Using Assumption A1, we find that $h^{-d/2}\{\bar{\theta}%
_{n}(c_{n,1},c_{n,2})-a_{n}^{R}(c_{n,1},c_{n,2})\}$ is equal to%
\begin{equation*}
h^{-d/2}\sum_{A\in \mathcal{N}_{J}}\{\zeta _{n,A}(B_{n,A}(c_{n,1},c_{n,2}))-%
\mathbf{E}\zeta _{N,A}(B_{n,A}(c_{n,1},c_{n,2}))\}+o_{P}(1),
\end{equation*}%
where for any Borel set $B\subset \mathcal{S}$,%
\begin{eqnarray*}
\zeta _{n,A}(B) &\equiv &\int_{B}\Lambda _{A,p}(\sqrt{nh^{d}}\mathbf{z}%
_{n,\tau }(x))dQ(x,\tau ), \\
\zeta _{N,A}(B) &\equiv &\int_{B}\Lambda _{A,p}(\sqrt{nh^{d}}\mathbf{z}%
_{N,\tau }(x))dQ(x,\tau ),
\end{eqnarray*}%
and%
\begin{equation*}
\mathbf{z}_{n,\tau }(x)\equiv \frac{1}{nh^{d}}\sum_{i=1}^{n}\beta _{n,x,\tau
}(Y_{i},(X_{i}-x)/h)-\frac{1}{h^{d}}\mathbf{E}\left[ \beta _{n,x,\tau
}(Y_{i},(X_{i}-x)/h)\right] \text{,}
\end{equation*}%
with 
\begin{equation*}
\beta _{n,x,\tau }(Y_{i},(X_{i}-x)/h)=(\beta _{n,x,\tau
,1}(Y_{i1},(X_{i}-x)/h),...,\beta _{n,x,\tau
,J}(Y_{iJ},(X_{i}-x)/h))^{\top }.
\end{equation*}%
We take $0<\bar{\varepsilon}_{l}\rightarrow 0$ as $l\rightarrow \infty $ and
take $\mathcal{C}_{l}\subset \mathbf{R}^{d}$ such that%
\begin{equation*}
0<P\left\{ X_{i}\in \mathbf{R}^{d}\backslash \mathcal{C}_{l}\right\} \leq 
\bar{\varepsilon}_{l},
\end{equation*}%
and $Q((\mathcal{X}\backslash \mathcal{C}_{l})\times \mathcal{T})\rightarrow
0$ as $l\rightarrow \infty $. Such a sequence $\{\bar{\varepsilon}%
_{l}\}_{l=1}^{\infty }$ exists by Assumption A6(ii) by the condition that $%
\mathcal{S}$ is compact. We write%
\begin{eqnarray}
&&\frac{h^{-d/2}\sum_{A\in \mathcal{N}_{J}}\{\zeta
_{n,A}(B_{n,A}(c_{n,1},c_{n,2}))-\mathbf{E}\zeta
_{N,A}(B_{n,A}(c_{n,1},c_{n,2}))\}}{\sigma _{n}^{2}(c_{n,1},c_{n,2})}
\label{dec56} \\
&=&\frac{h^{-d/2}\sum_{A\in \mathcal{N}_{J}}\{\zeta
_{n,A}(B_{n,A}(c_{n,1},c_{n,2})\cap (\mathcal{C}_{l}\times \mathcal{T}))-%
\mathbf{E}\zeta _{N,A}(B_{n,A}(c_{n,1},c_{n,2})\cap (\mathcal{C}_{l}\times 
\mathcal{T}))\}}{\sigma _{n}^{2}(c_{n,1},c_{n,2})}  \notag \\
&&+\frac{h^{-d/2}\sum_{A\in \mathcal{N}_{J}}\{\zeta
_{n,A}(B_{n,A}(c_{n,1},c_{n,2})\backslash (\mathcal{C}_{l}\times \mathcal{T}%
))-\mathbf{E}\zeta _{N,A}(B_{n,A}(c_{n,1},c_{n,2})\backslash (\mathcal{C}%
_{l}\times \mathcal{T}))\}}{\sigma _{n}^{2}(c_{n,1},c_{n,2})}  \notag \\
&=&A_{1n}+A_{2n}\text{, say.}  \notag
\end{eqnarray}%
As for $A_{2n}$, we apply Lemma B7 in Appendix B, and the condition that $Q((%
\mathcal{X}\backslash \mathcal{C}_{l})\times \mathcal{T})\rightarrow 0$, as $%
l\rightarrow \infty $, and 
\begin{equation*}
\text{liminf}_{n\rightarrow \infty }\text{inf}_{P\in \mathcal{P}_{0}}\sigma
_{n}(c_{1n},c_{2n})>0\text{,}
\end{equation*}%
to deduce that $A_{2n}=o_{P}(1)$, as $n\rightarrow \infty $ and then $%
l\rightarrow \infty $. As for $A_{1n}$, first observe that as $n\rightarrow
\infty $ and then $l\rightarrow \infty ,$%
\begin{equation}
\left\vert \sigma _{n}^{2}(c_{n,1},c_{n,2})-\bar{\sigma}%
_{n,l}^{2}(c_{n,1},c_{n,2})\right\vert \rightarrow 0,  \label{conv}
\end{equation}%
where $\bar{\sigma}_{n,l}^{2}(c_{n,1},c_{n,2})$ is equal to $\sigma
_{n}^{2}(c_{n,1},c_{n,2})$ except that $B_{n,A}(c_{n,1},c_{n,2})$'s are
replaced by $B_{n,A}(c_{n,1},c_{n,2})\cap (\mathcal{C}_{l}\times \mathcal{T}%
) $. The convergence follows by Assumption 6(i). Also by Lemma B9(i) and the
convergence in (\ref{conv}) and the fact that 
\begin{equation*}
\liminf_{n\rightarrow \infty }\inf_{P\in \mathcal{P}_{0}}\sigma
_{n}^{2}(c_{n,1},c_{n,2})>0,
\end{equation*}%
we have%
\begin{equation*}
A_{1n}\overset{d}{\rightarrow }N(0,1),\text{ }\mathcal{P}_{0}\text{%
-uniformly,}
\end{equation*}%
as $n\rightarrow \infty $ and as $l\rightarrow \infty $. Hence we obtain (i).

\noindent (ii) The proof can be done in the same way as in the proof of (i),
using Lemmas C7 and C9(i) in Appendix C instead of Lemmas B7 and B9(i) in
Appendix B.
\end{proof}

\begin{LemmaA}
\textit{Suppose that Assumptions A\ref{assumption-A1}-A\ref{assumption-A5}
hold. Then for any sequences }$c_{n,L},c_{n,U}>0$\textit{\ satisfying
Assumption A4(ii), and for each} $A\in \mathcal{N}_{J},$%
\begin{equation*}
\inf_{P\in \mathcal{P}}P\left\{ B_{n,A}(c_{n,L},c_{n,U})\subset \hat{B}_{A}(%
\hat{c}_{n})\subset B_{n,A}(c_{n,U},c_{n,L})\right\} \rightarrow 1\text{%
\textit{, as} }n\rightarrow \infty \text{.}
\end{equation*}
\end{LemmaA}

\begin{proof}[Proof of Lemma A3]
By using Assumptions A\ref{assumption-A3}-A\ref{assumption-A5}, and
following the proof of Theorem 2, Claim 1 in %
\citeasnoun{Linton/Song/Whang:10}, we can complete the proof. Details are
omitted.
\end{proof}

Define for $c_{n,1},c_{n,2}>0,$%
\begin{eqnarray*}
T_{n}(c_{n,1},c_{n,2}) &\equiv &h^{-d/2}\left( \frac{\bar{\theta}%
_{n}(c_{n,1},c_{n,2})-a_{n}(c_{n,1},c_{n,2})}{\sigma _{n}(c_{n,1},c_{n,2})}%
\right) \text{ and} \\
T_{n}^{\ast }(c_{n,1},c_{n,2}) &\equiv &h^{-d/2}\left( \frac{\bar{\theta}%
_{n}^{\ast }(c_{n,1},c_{n,2})-a_{n}(c_{n,1},c_{n,2})}{\sigma
_{n}(c_{n,1},c_{n,2})}\right) .
\end{eqnarray*}%
We introduce critical values for the finite sample distribution of $\hat{%
\theta}$ as follows:%
\begin{equation*}
\gamma _{n}^{\alpha }(c_{n,1},c_{n,2})\equiv \inf \left\{ c\in \mathbf{R}%
:P\left\{ T_{n}(c_{n,1},c_{n,2})\leq c\right\} >1-\alpha \right\} \text{.}
\end{equation*}%
Similarly, let us introduce bootstrap critical values:%
\begin{equation}
\gamma _{n}^{\alpha \ast }(c_{n,1},c_{n,2})\equiv \inf \left\{ c\in \mathbf{R%
}:P^{\ast }\left\{ T_{n}^{\ast }(c_{n,1},c_{n,2})\leq c\right\} >1-\alpha
\right\} \text{.}  \label{c_tildee*}
\end{equation}%
Finally, we introduce asymptotic critical values:\ $\gamma _{\infty
}^{\alpha }\equiv \Phi ^{-1}(1-\alpha ),$ where $\Phi $ denotes the standard
normal CDF.

\begin{LemmaA}
Suppose that Assumptions A\ref{assumption-A1}-A\ref{assumption-A3}, A4(i)%
\textit{, and }A5-A6 hold. Then the following holds.

\noindent (i)\textit{\ }For any $c_{n,1},c_{n,2}\rightarrow \infty $ such
that 
\begin{equation*}
\liminf_{n\rightarrow \infty }\inf_{P\in \mathcal{P}}\sigma
_{n}^{2}(c_{n,1},c_{n,2})>0,
\end{equation*}%
it is satisfied that%
\begin{equation*}
\sup_{P\in \mathcal{P}}\left\vert \gamma _{n}^{\alpha
}(c_{n,1},c_{n,2})-\gamma _{\infty }^{\alpha }\right\vert \rightarrow 0,\ 
\text{as\ }n\rightarrow \infty .
\end{equation*}

\noindent (ii) \textit{Suppose further that Assumptions B\ref{assumption-B1}
and B\ref{assumption-B4} hold. Then }for any $c_{n,1},c_{n,2}\rightarrow
\infty $ such that 
\begin{equation*}
\liminf_{n\rightarrow \infty }\inf_{P\in \mathcal{P}}\sigma
_{n}^{2}(c_{n,1},c_{n,2})>0,
\end{equation*}%
it is satisfied that%
\begin{equation*}
\sup_{P\in \mathcal{P}}\left\vert \gamma _{n}^{\alpha \ast
}(c_{n,1},c_{n,2})-\gamma _{\infty }^{\alpha }\right\vert \rightarrow 0,\ 
\text{as\ }n\rightarrow \infty .
\end{equation*}
\end{LemmaA}

\begin{proof}[Proof of Lemma A4]
(i) The statement immediately follows from the first statement of Lemma
A2(i) and Lemma A1.

\noindent (ii) We show only the second statement. Fix $a>0$. Let us
introduce two events:%
\begin{equation*}
E_{n,1}\equiv \left\{ \gamma _{n}^{\alpha \ast }(c_{n,1},c_{n,2})-\gamma
_{\infty }^{\alpha }<-a\right\} \text{ and }E_{n,2}\equiv \left\{ \gamma
_{n}^{\alpha \ast }(c_{n,1},c_{n,2})-\gamma _{\infty }^{\alpha }>a\right\} .
\end{equation*}%
On the event $E_{n,1},$ we have%
\begin{eqnarray*}
\alpha &=&P^{\ast }\left\{ h^{-d/2}\left( \frac{\bar{\theta}_{n}^{\ast
}(c_{n,1},c_{n,2})-a_{n}(c_{n,1},c_{n,2})}{\sigma _{n}(c_{n,1},c_{n,2})}%
\right) >\gamma _{n}^{\alpha \ast }(c_{n,1},c_{n,2})\right\} \\
&\geq &P^{\ast }\left\{ h^{-d/2}\left( \frac{\bar{\theta}_{n}^{\ast
}(c_{n,1},c_{n,2})-a_{n}(c_{n,1},c_{n,2})}{\sigma _{n}(c_{n,1},c_{n,2})}%
\right) >\gamma _{\infty }^{\alpha }-a\right\} .
\end{eqnarray*}%
By Lemma A2(ii) and Lemma A1, the last probability is equal to%
\begin{equation*}
1-\Phi \left( \gamma _{\infty }^{\alpha }-a\right) +o_{P}(1)>\alpha
+o_{P}(1),
\end{equation*}%
where $o_{P}(1)$ is uniform over $P\in \mathcal{P}$ and the last strict
inequality follows by the definition of $\gamma _{\infty }^{\alpha }$ and $%
a>0.$ Hence $\sup_{P\in \mathcal{P}}PE_{n,1}\rightarrow 0$ as $n\rightarrow
\infty $. Similarly, on the event $E_{n,2}$, we have%
\begin{eqnarray*}
\alpha &=&P^{\ast }\left\{ h^{-d/2}\left( \frac{\bar{\theta}_{n}^{\ast
}(c_{n,1},c_{n,2})-a_{n}(c_{n,1},c_{n,2})}{\sigma _{n}(c_{n,1},c_{n,2})}%
\right) >\gamma _{n}^{\alpha \ast }(c_{n,1},c_{n,2})\right\} \\
&\leq &P^{\ast }\left\{ h^{-d/2}\left( \frac{\bar{\theta}_{n}^{\ast
}(c_{n,1},c_{n,2})-a_{n}(c_{n,1},c_{n,2})}{\sigma _{n}(c_{n,1},c_{n,2})}%
\right) >\gamma _{\infty }^{\alpha }+a\right\} .
\end{eqnarray*}%
By the first statement of Lemma A2(ii) and Lemma A1, the last bootstrap
probability is bounded by%
\begin{equation*}
1-\Phi \left( \gamma _{\infty }^{\alpha }+a\right) +o_{P}(1)<\alpha
+o_{P}(1),
\end{equation*}%
so that we have $\sup_{P\in \mathcal{P}}PE_{n,2}\rightarrow 0$ as $%
n\rightarrow \infty $.\textbf{\ }We conclude that%
\begin{equation*}
\sup_{P\in \mathcal{P}}P\left\{ |\gamma _{n}^{\alpha \ast
}(c_{n,1},c_{n,2})-\gamma _{\infty }^{\alpha }|>a\right\} =\sup_{P\in 
\mathcal{P}}\left( PE_{n,1}+PE_{n,2}\right) \rightarrow 0,
\end{equation*}%
as $n\rightarrow \infty $, obtaining the desired result.
\end{proof}

\begin{proof}[Proof of Theorem \protect\ref{Thm1}]
By Lemma 1, we have 
\begin{equation*}
\inf_{P\in \mathcal{P}_{0}}P\left\{ \hat{\theta}=\sum_{A\in \mathcal{N}%
_{J}}\int_{B_{n,A}(c_{n,L},c_{n,U})}\Lambda _{A,p}\left( \mathbf{\hat{u}}%
_{\tau }(x)\right) dQ(x,\tau )\right\} \rightarrow 1,
\end{equation*}%
as $n\rightarrow \infty $. Since under the null hypothesis, we have $%
v_{n,\tau ,j}(\cdot )/\hat{\sigma}_{\tau ,j}(\cdot )\leq 0$ for all $j\in 
\mathbb{N}_{J},$ with probability approaching one by Assumption A5, we have with probability approaching one (uniformly over $P \in \mathcal{P}_0$),
\begin{eqnarray*}
&&\sum_{A\in \mathcal{N}_{J}}\int_{B_{n,A}(c_{n,L},c_{n,U})}\Lambda
_{A,p}\left( \mathbf{\hat{u}}_{\tau }(x)\right) dQ(x,\tau ) \\
&\leq &\sum_{A\in \mathcal{N}_{J}}\int_{B_{n,A}(c_{n,L},c_{n,U})}\Lambda
_{A,p}\left( \mathbf{\hat{s}}_{\tau }(x)\right) dQ(x,\tau )\equiv \bar{\theta%
}_{n}(c_{n,L},c_{n,U}).
\end{eqnarray*}%
Thus, we have as $n\rightarrow \infty ,$%
\begin{equation}
\inf_{P\in \mathcal{P}_{0}}P\left\{ \hat{\theta}\leq \bar{\theta}%
_{n}(c_{n,L},c_{n,U})\right\} \rightarrow 1.  \label{ineq42b}
\end{equation}

Let the $(1-\alpha )$-th percentile of the bootstrap distribution of%
\begin{equation*}
\bar{\theta}_{n}^{\ast }(c_{n,L},c_{n,U})=\sum_{A\in \mathcal{N}%
_{J}}\int_{B_{n,A}(c_{n,L},c_{n,U})}\Lambda _{A,p}(\mathbf{\hat{s}}_{\tau
}^{\ast }(x))dQ(x,\tau )
\end{equation*}%
be denoted by $\bar{c}_{n,L}^{\alpha \ast }$. By Lemma A3 and Assumption
A4(ii), with probability approaching one,%
\begin{equation}
\sum_{A\in \mathcal{N}_{J}}\int_{B_{n,A}(c_{n,L},c_{n,U})}\Lambda
_{A,p}\left( \mathbf{\hat{s}}_{\tau }^{\ast }(x)\right) dQ(x,\tau )\leq
\sum_{A\in \mathcal{N}_{J}}\int_{\hat{B}_{A}(\hat{c}_{n})}\Lambda
_{A,p}\left( \mathbf{\hat{s}}_{\tau }^{\ast }(x)\right) dQ(x,\tau ).
\label{comp}
\end{equation}%
This implies that as $n\rightarrow \infty ,$%
\begin{equation}
\inf_{P\in \mathcal{P}}P\left\{ c_{\alpha }^{\ast }\geq \bar{c}%
_{n,L}^{\alpha \ast }\right\} \rightarrow 1\text{.}  \label{comp2}
\end{equation}%
There exists a sequence of probabilities $\{P_{n}\}_{n\geq 1}\subset 
\mathcal{P}_{0}$ such that 
\begin{eqnarray}
\underset{n\rightarrow \infty }{\text{limsup}}\sup_{P\in \mathcal{P}%
_{0}}P\left\{ \hat{\theta}>c_{\alpha ,\eta }^{\ast }\right\} &=&\ \underset{%
n\rightarrow \infty }{\text{limsup}}P_{n}\left\{ \hat{\theta}>c_{\alpha
,\eta }^{\ast }\right\}  \label{limit} \\
&=&\text{lim}_{n\rightarrow \infty }P_{w_{n}}\left\{ \hat{\theta}%
_{w_{n}}>c_{w_{n},\alpha ,\eta }^{\ast }\right\} ,  \notag
\end{eqnarray}%
where $\{w_{n}\}\subset \{n\}$ is a certain subsequence, and $\hat{\theta}%
_{w_{n}}$ and $c_{w_{n},\alpha ,\eta }^{\ast }$ are the same as $\hat{\theta}
$ and $c_{\alpha ,\eta }^{\ast }$ except that the sample size $n$ is now
replaced by $w_{n}$.\newline
By Assumption A6(i), $\{\sigma _{n}(c_{n,L},c_{n,U})\}_{n\geq 1}$ is a
bounded sequence. Therefore, there exists a subsequence $\{u_{n}\}_{n\geq
1}\subset \{w_{n}\}_{n\geq 1},$ such that $\sigma
_{u_{n}}(c_{u_{n},L},c_{u_{n},U})$ converges. We consider two cases:

\textbf{Case 1: }lim$_{n\rightarrow \infty }\sigma
_{u_{n}}(c_{u_{n},L},c_{u_{n},U})>0,$ and

\textbf{Case 2:} lim$_{n\rightarrow \infty }\sigma
_{u_{n}}(c_{u_{n},L},c_{u_{n},U})=0.$

In both cases, we will show below that 
\begin{equation}
\underset{n\rightarrow \infty }{\text{limsup}}P_{u_{n}}\{\hat{\theta}%
_{u_{n}}>c_{u_{n},\alpha ,\eta }^{\ast }\}\leq \alpha .  \label{control}
\end{equation}%
Since along $\{w_{n}\}$, $P_{w_{n}}\{\hat{\theta}_{w_{n}}>c_{w_{n},\alpha
,\eta }^{\ast }\}$ converges, it does so along any subsequence of $\{w_{n}\}$%
. Therefore, the above limsup is equal to the last limit in (\ref{limit}).
This completes the proof.\newline
\textbf{Proof of (\ref{control}) in Case 1: }We write $P_{u_{n}}\{\hat{\theta%
}_{u_{n}}>c_{u_{n},\alpha ,\eta }^{\ast }\}$ as%
\begin{eqnarray*}
&&P_{u_{n}}\left( h^{-d/2}\left( \frac{\hat{\theta}%
_{u_{n}}-a_{u_{n}}(c_{u_{n},L},c_{u_{n},U})}{\sigma
_{u_{n}}(c_{u_{n},L},c_{u_{n},U})}\right) >h^{-d/2}\left( \frac{%
c_{u_{n},\alpha ,\eta }^{\ast }-a_{u_{n}}(c_{u_{n},L},c_{u_{n},U})}{\sigma
_{u_{n}}(c_{u_{n},L},c_{u_{n},U})}\right) \right) \\
&\leq &P_{u_{n}}\left( h^{-d/2}\left( \frac{\hat{\theta}%
_{u_{n}}-a_{u_{n}}(c_{u_{n},L},c_{u_{n},U})}{\sigma
_{u_{n}}(c_{u_{n},L},c_{u_{n},U})}\right) >h^{-d/2}\left( \frac{\bar{c}%
_{u_{n},L}^{\alpha \ast }-a_{u_{n}}(c_{u_{n},L},c_{u_{n},U})}{\sigma
_{u_{n}}(c_{u_{n},L},c_{u_{n},U})}\right) \right) +o(1),
\end{eqnarray*}%
where the inequality follows by the fact that $c_{\alpha ,\eta }^{\ast }\geq
c_{\alpha }^{\ast }\geq \bar{c}_{n,L}^{\alpha \ast }$ with probability
approaching one by (\ref{comp2}). Using (\ref{ineq42b}), we bound the last
probability by 
\begin{equation}
P_{u_{n}}\left\{ h^{-d/2}\left( \frac{\bar{\theta}%
_{u_{n}}(c_{u_{n},L},c_{u_{n},U})-a_{u_{n}}(c_{u_{n},L},c_{u_{n},U})}{\sigma
_{u_{n}}(c_{u_{n},L},c_{u_{n},U})}\right) >h^{-d/2}\left( \frac{\bar{c}%
_{u_{n},L}^{\alpha \ast }-a_{u_{n}}(c_{u_{n},L},c_{u_{n},U})}{\sigma
_{u_{n}}(c_{u_{n},L},c_{u_{n},U})}\right) \right\} +o(1).  \label{probs}
\end{equation}%
Therefore, since lim$_{n\rightarrow \infty }\sigma
_{u_{n}}(c_{u_{n},L},c_{u_{n},U})>0$, by Lemmas A2 and A4, we rewrite the
last probability in (\ref{probs}) as%
\begin{eqnarray*}
&&P_{u_{n}}\left\{ h^{-d/2}\left( \frac{\bar{\theta}%
_{u_{n}}(c_{u_{n},L},c_{u_{n},U})-a_{u_{n}}(c_{u_{n},L},c_{u_{n},U})}{\sigma
_{u_{n}}(c_{u_{n},L},c_{u_{n},U})}\right) >\gamma _{u_{n}}^{\alpha \ast
}(c_{u_{n},L},c_{u_{n},U})\right\} +o(1) \\
&=&P_{u_{n}}\left\{ h^{-d/2}\left( \frac{\bar{\theta}%
_{u_{n}}(c_{u_{n},L},c_{u_{n},U})-a_{u_{n}}(c_{u_{n},L},c_{u_{n},U})}{\sigma
_{u_{n}}(c_{u_{n},L},c_{u_{n},U})}\right) >\gamma _{\infty }^{\alpha
}\right\} +o(1)=\alpha +o(1).
\end{eqnarray*}%
This completes the proof of Step 1.\bigskip \newline
\textbf{Proof of (\ref{control}) in Case 2:} First, observe that%
\begin{equation*}
a_{u_{n}}^{\ast }(c_{u_{n},L},c_{u_{n},U})\leq a_{u_{n}}^{\ast }(\hat{c}%
_{u_{n}}),
\end{equation*}%
with probability approaching one by Lemma A3. Hence using this and (\ref%
{ineq42b}),%
\begin{eqnarray*}
P_{u_{n}}\left\{ \hat{\theta}_{u_{n}}>c_{u_{n},\alpha ,\eta }^{\ast
}\right\} &=&P_{u_{n}}\left\{ h^{-d/2}\left( \hat{\theta}%
_{u_{n}}-a_{u_{n}}(c_{u_{n},L},c_{u_{n},U})\right) >h^{-d/2}\left(
c_{u_{n},\alpha ,\eta }^{\ast }-a_{u_{n}}(c_{u_{n},L},c_{u_{n},U})\right)
\right\} \\
&\leq &P_{u_{n}}\left\{ 
\begin{array}{c}
h^{-d/2}\left( \bar{\theta}%
_{u_{n}}(c_{u_{n},L},c_{u_{n},U})-a_{u_{n}}(c_{u_{n},L},c_{u_{n},U})\right)
\\ 
>h^{-d/2}\left( h^{d/2}\eta +a_{u_{n}}^{\ast
}(c_{u_{n},L},c_{u_{n},U})-a_{u_{n}}(c_{u_{n},L},c_{u_{n},U})\right)%
\end{array}%
\right\} +o(1).
\end{eqnarray*}%
By Lemma A1, the leading probability is equal to%
\begin{equation*}
P_{u_{n}}\left\{ h^{-d/2}\left( \bar{\theta}%
_{u_{n}}(c_{u_{n},L},c_{u_{n},U})-a_{u_{n}}(c_{u_{n},L},c_{u_{n},U})\right)
>\eta +o_{P}(1)\right\} +o(1).
\end{equation*}%
Since $\eta >0$ and lim$_{n\rightarrow \infty }\sigma
_{u_{n}}(c_{u_{n},L},c_{u_{n},U})=0$, the leading probability vanishes by
Lemma B9(ii).
\end{proof}

\begin{proof}[Proof of Theorem \protect\ref{Thm2}]
We focus on probabilities $P\in \mathcal{P}_{n}(\lambda _{n},q_{n})\cap 
\mathcal{P}_{0}$. Recalling the definition of $\mathbf{u}_{n,\tau }(x;\hat{%
\sigma})\equiv \left[ r_{n,j}v_{n,\tau ,j}(x)/\hat{\sigma}_{\tau ,j}(x)%
\right] _{j\in \mathbb{N}_{J}}$ and applying Lemma 1 along with the
condition that%
\begin{equation*}
\sqrt{\log n}/c_{n,U}<\sqrt{\log n}/c_{n,L}\rightarrow 0,
\end{equation*}%
as $n\rightarrow \infty $, we find that with probability approaching one,%
\begin{eqnarray*}
\hat{\theta} &=&\sum_{A\in \mathcal{N}_{J}}\int_{B_{n,A}(c_{n,U},c_{n,L})}%
\Lambda _{A,p}\left( \mathbf{\hat{s}}_{\tau }(x)+\mathbf{u}_{n,\tau }(x;\hat{%
\sigma})\right) dQ(x,\tau ) \\
&=&\sum_{A\in \mathcal{N}_{J}}\int_{B_{n,A}(q_{n})}\Lambda _{A,p}\left( 
\mathbf{\hat{s}}_{\tau }(x)+\mathbf{u}_{n,\tau }(x;\hat{\sigma})\right)
dQ(x,\tau ) \\
&&+\sum_{A\in \mathcal{N}_{J}}\int_{B_{n,A}(c_{n,U},c_{n,L})\backslash
B_{n,A}(q_{n})}\Lambda _{A,p}\left( \mathbf{\hat{s}}_{\tau }(x)+\mathbf{u}%
_{n,\tau }(x;\hat{\sigma})\right) dQ(x,\tau ).
\end{eqnarray*}%
Since under $P\in \mathcal{P}_{0}$, $\mathbf{u}_{n,\tau }(x;\hat{\sigma}%
)\leq 0$ for all $x\in \mathcal{S}$, with probability approaching one by
Assumption 5, the last term multiplied by $h^{-d/2}$ is bounded by (from
some large $n$ on)%
\begin{eqnarray*}
&&h^{-d/2}\sum_{A\in \mathcal{N}_{J}}\int_{B_{n,A}(c_{n,U},c_{n,L})%
\backslash B_{n,A}(q_{n})}\Lambda _{A,p}\left( \mathbf{\hat{s}}_{\tau
}(x)\right) dQ(x,\tau ) \\
&\leq &h^{-d/2}\sum_{A\in \mathcal{N}_{J}}\left( \sup_{(x,\tau )\in \mathcal{%
S}}||\mathbf{\hat{s}}_{\tau }(x)||\right) ^{p}Q\left(
B_{n,A}(c_{n,U},c_{n,L})\backslash B_{n,A}(q_{n})\right) \\
&=&O_{P}\left( h^{-d/2}(\log n)^{p/2}\lambda _{n}\right) =o_{P}(1),
\end{eqnarray*}%
where the second to last equality follows because $Q\left(
B_{n,A}(c_{n,U},c_{n,L})\backslash B_{n,A}(q_{n})\right) \leq \lambda _{n}$
by the definition of $\mathcal{P}_{n}(\lambda _{n},q_{n})$, and the last
equality follows by (\ref{lamq}).

On the other hand,%
\begin{eqnarray*}
&&h^{-d/2}\sum_{A\in \mathcal{N}_{J}}\int_{B_{n,A}(q_{n})}\Lambda
_{A,p}\left( \mathbf{\hat{s}}_{\tau }(x)+\mathbf{u}_{n,\tau }(x;\hat{\sigma}%
)\right) dQ(x,\tau ) \\
&=&h^{-d/2}\sum_{A\in \mathcal{N}_{J}}\int_{B_{n,A}(q_{n})}\Lambda
_{A,p}\left( \mathbf{\hat{s}}_{\tau }(x)\right) dQ(x,\tau ) \\
&&+h^{-d/2}\sum_{A\in \mathcal{N}_{J}}\int_{B_{n,A}(q_{n})}\Lambda
_{A,p}\left( \mathbf{\hat{s}}_{\tau }(x)+\mathbf{u}_{n,\tau }(x;\hat{\sigma}%
)\right) dQ(x,\tau ) \\
&&-h^{-d/2}\sum_{A\in \mathcal{N}_{J}}\int_{B_{n,A}(q_{n})}\Lambda
_{A,p}\left( \mathbf{\hat{s}}_{\tau }(x)\right) dQ(x,\tau ).
\end{eqnarray*}%
From the definition of $\Lambda _{p}$ in (\ref{lambda_p}), the last
difference (in absolute value) is bounded by%
\begin{eqnarray*}
&&Ch^{-d/2}\sum_{A\in \mathcal{N}_{J}}\int_{B_{n,A}(q_{n})}\left\Vert [%
\mathbf{u}_{n,\tau }(x;\hat{\sigma})]_{A}\right\Vert \left\Vert [\mathbf{%
\hat{s}}_{\tau }(x)]_{A}\right\Vert ^{p-1}dQ(x,\tau ) \\
&&+Ch^{-d/2}\sum_{A\in \mathcal{N}_{J}}\int_{B_{n,A}(q_{n})}\left\Vert [%
\mathbf{u}_{n,\tau }(x;\hat{\sigma})]_{A}\right\Vert \left\Vert [\mathbf{u}%
_{n,\tau }(x;\hat{\sigma})]_{A}\right\Vert ^{p-1}dQ(x,\tau ),
\end{eqnarray*}%
where $[a]_{A}$ is a vector $a$ with the $j$-th entry is set to zero for
all $j\in \mathbb{N}_{J}\backslash A$ and $C>0$ is a constant that does not
depend on $n\geq 1$ or $P\in \mathcal{P}$. We have sup$_{(x,\tau )\in
B_{n,A}(q_{n})}\left\Vert [\mathbf{u}_{n,\tau }(x;\hat{\sigma}%
)]_{A}\right\Vert \leq q_{n}(1+o_{P}(1))$, by the null hypothesis and by
Assumption A5. Also, by Assumptions A3 and A5, 
\begin{equation*}
\text{sup}_{(x,\tau )\in B_{n,A}(q_{n})}\left\Vert [\mathbf{\hat{s}}_{\tau
}(x)]_{A}\right\Vert =O_{P}\left( \sqrt{\log n}\right) .
\end{equation*}%
Therefore, we conclude that%
\begin{eqnarray*}
&&h^{-d/2}\sum_{A\in \mathcal{N}_{J}}\int_{B_{n,A}(q_{n})}\Lambda
_{A,p}\left( \mathbf{\hat{s}}_{\tau }(x)+\mathbf{u}_{n,\tau }(x;\hat{\sigma}%
)\right) dQ(x,\tau ) \\
&=&h^{-d/2}\sum_{A\in \mathcal{N}_{J}}\int_{B_{n,A}(q_{n})}\Lambda
_{A,p}\left( \mathbf{\hat{s}}_{\tau }(x)\right) dQ(x,\tau )+O_{P}\left(
h^{-d/2}q_{n}\{(\log n)^{(p-1)/2}+q_{n}^{p-1}\}\right) .
\end{eqnarray*}%
The last $O_{P}(1)$ term is $o_{P}(1)$ by the condition for $q_{n}$ in (\ref%
{lamq}). Thus we find that%
\begin{equation}
\hat{\theta}=\bar{\theta}_{n}(q_{n})+o_{P}(h^{d/2}),  \label{cv33}
\end{equation}%
where $\bar{\theta}_{n}(q_{n})=\sum_{A\in \mathcal{N}_{J}}%
\int_{B_{n,A}(q_{n})}\Lambda _{A,p}\left( \mathbf{\hat{s}}_{\tau }(x)\right)
dQ(x,\tau ).$

Now let us consider the bootstrap statistic. We write%
\begin{eqnarray*}
\hat{\theta}^{\ast } &=&\sum_{A\in \mathcal{N}_{J}}\int_{\hat{B}_{A}(\hat{c}%
_{n})}\Lambda _{A,p}\left( \mathbf{\hat{s}}_{\tau }^{\ast }(x)\right)
dQ(x,\tau ) \\
&=&\sum_{A\in \mathcal{N}_{J}}\int_{B_{n,A}(q_{n})}\Lambda _{A,p}\left( 
\mathbf{\hat{s}}_{\tau }^{\ast }(x)\right) dQ(x,\tau )+\sum_{A\in \mathcal{N}%
_{J}}\int_{\hat{B}_{A}(\hat{c}_{n})\backslash B_{n,A}(q_{n})}\Lambda
_{A,p}\left( \mathbf{\hat{s}}_{\tau }^{\ast }(x)\right) dQ(x,\tau ).
\end{eqnarray*}%
By Lemma A3, we find that 
\begin{equation*}
\inf_{P\in \mathcal{P}}P\left\{ \hat{B}_{n,A}(\hat{c}_{n})\subset
B_{n,A}(c_{n,U},c_{n,L})\right\} \rightarrow 1,\text{ as }n\rightarrow
\infty ,
\end{equation*}%
so that%
\begin{equation*}
\sum_{A\in \mathcal{N}_{J}}\int_{\hat{B}_{A}(\hat{c}_{n})\backslash
B_{n,A}(q_{n})}\Lambda _{A,p}\left( \mathbf{\hat{s}}_{\tau }^{\ast
}(x)\right) dQ(x,\tau )\leq \sum_{A\in \mathcal{N}_{J}}%
\int_{B_{n,A}(c_{n,U},c_{n,L})\backslash B_{n,A}(q_{n})}\Lambda _{A,p}\left( 
\mathbf{\hat{s}}_{\tau }^{\ast }(x)\right) dQ(x,\tau ),
\end{equation*}%
with probability approaching one. The last term multiplied by $h^{-d/2}$ is
bounded by%
\begin{eqnarray*}
&&h^{-d/2}\left( \sup_{(x,\tau )\in \mathcal{S}}||\mathbf{\hat{s}}_{\tau
}^{\ast }(x)||\right) ^{p}\sum_{A\in \mathcal{N}_{J}}Q\left(
B_{n,A}(c_{n,U},c_{n,L})\backslash B_{n,A}(q_{n})\right) \\
&=&O_{P^{\ast }}\left( h^{-d/2}(\log n)^{p/2}\lambda _{n}\right) =o_{P^{\ast
}}(1),\text{ }\mathcal{P}_{n}(\lambda _{n},q_{n})\text{-uniformly,}
\end{eqnarray*}%
where the second to last equality follows by Assumption B2 and the
definition of $\mathcal{P}_{n}(\lambda _{n},q_{n})$, and the last equality
follows by (\ref{lamq}). Thus, we conclude that%
\begin{equation}
\frac{h^{-d/2}(\hat{\theta}^{\ast }-a_{n}(q_{n}))}{\sigma _{n}(q_{n})}=\frac{%
h^{-d/2}\left( \bar{\theta}_{n}^{\ast }(q_{n})-a_{n}(q_{n})\right) }{\sigma
_{n}(q_{n})}+o_{P^{\ast }}(1),\text{ }\mathcal{P}_{n}(\lambda _{n},q_{n})%
\text{-uniformly,}  \label{cv22}
\end{equation}%
where%
\begin{equation*}
\bar{\theta}^{\ast }(q_{n})\equiv \sum_{A\in \mathcal{N}_{J}}%
\int_{B_{n,A}(q_{n})}\Lambda _{A,p}\left( \mathbf{\hat{s}}_{\tau }^{\ast
}(x)\right) dQ(x,\tau ).
\end{equation*}

Using the same arguments, we also observe that 
\begin{equation}
\hat{a}^{\ast }=\hat{a}^{\ast
}(q_{n})+o_{P}(h^{d/2})=a_{n}(q_{n})+o_{P}(h^{d/2}),  \label{a}
\end{equation}%
where the last equality uses Lemma A1. Let the $(1-\alpha )$-th percentile
of the bootstrap distribution of $\bar{\theta}^{\ast }(q_{n})$ be denoted by 
$\bar{c}_{n}^{\alpha \ast }(q_{n})$. Then by (\ref{cv22}), we have 
\begin{equation}
\frac{h^{-d/2}\left( c_{\alpha }^{\ast }-a_{n}(q_{n})\right) }{\sigma
_{n}(q_{n})}=\frac{h^{-d/2}\left( \bar{c}_{n}^{\alpha \ast
}(q_{n})-a_{n}(q_{n})\right) }{\sigma _{n}(q_{n})}+o_{P^{\ast }}(1),\text{ }%
\mathcal{P}_{n}(\lambda _{n},q_{n})\text{-uniformly.}  \label{cv7}
\end{equation}%
By Lemma A4(ii) and by the condition that $\sigma _{n}(q_{n})\geq \eta /\Phi
^{-1}(1-\alpha )$, the leading term on the right hand side is equal to%
\begin{equation*}
\Phi ^{-1}(1-\alpha )+o_{P^{\ast }}(1)\text{, }\mathcal{P}_{n}(\lambda
_{n},q_{n})\text{-uniformly.}
\end{equation*}%
Note that 
\begin{equation}
c_{\alpha }^{\ast }\geq h^{d/2}\eta +\hat{a}_{n}^{\ast }+o_{P}(h^{d/2}),
\label{c_ehta}
\end{equation}%
by the restriction $\sigma _{n}(q_{n})\geq \eta /\Phi ^{-1}(1-\alpha )$ in
the definition of $\mathcal{P}_{n}(\lambda _{n},q_{n})$ and (\ref{a}). Using
this, and following the proof of Step 1 in the proof of Theorem 2, we deduce
that%
\begin{eqnarray*}
&&P\left\{ h^{-d/2}\left( \frac{\hat{\theta}-a_{n}(q_{n})}{\sigma _{n}(q_{n})%
}\right) >h^{-d/2}\left( \frac{c_{\alpha ,\eta }^{\ast }-a_{n}(q_{n})}{%
\sigma _{n}(q_{n})}\right) \right\} \\
&=&P\left\{ h^{-d/2}\left( \frac{\bar{\theta}_{n}(q_{n})-a_{n}(q_{n})}{%
\sigma _{n}(q_{n})}\right) >h^{-d/2}\left( \frac{c_{\alpha }^{\ast
}-a_{n}(q_{n})}{\sigma _{n}(q_{n})}\right) \right\} +o(1) \\
&=&P\left\{ h^{-d/2}\left( \frac{\bar{\theta}_{n}(q_{n})-a_{n}(q_{n})}{%
\sigma _{n}(q_{n})}\right) >h^{-d/2}\left( \frac{\bar{c}_{n}^{\alpha \ast
}(q_{n})-a_{n}(q_{n})}{\sigma _{n}(q_{n})}\right) \right\} +o(1),
\end{eqnarray*}%
where the first equality uses (\ref{cv33}) and (\ref{c_ehta}), and the second
equality uses (\ref{cv7}). Since $\sigma _{n}(q_{n})\geq \eta /\Phi
^{-1}(1-\alpha )>0$ for all $P\in \mathcal{P}_{n}(\lambda _{n},q_{n})\cap 
\mathcal{P}_{0}$ by definition, using the same arguments in the proof of
Lemma A4, we obtain that the last probability is equal to%
\begin{equation*}
\alpha +o(1),
\end{equation*}%
uniformly over $P\in \mathcal{P}_{n}(\lambda _{n},q_{n})\cap \mathcal{P}_{0}$%
.
\end{proof}

\begin{proof}[Proof of Theorem \protect\ref{Thm3}]
For any convex nonnegative map $f$ on $\mathbf{R}^{J},$ we have $2f(b/2)\leq
f(a+b)+f(-a)$. Hence we find that%
\begin{eqnarray*}
\hat{\theta} &=&\int \Lambda _{p}\left( \mathbf{\hat{s}}_{\tau }(x)+\mathbf{u%
}_{\tau }(x;\hat{\sigma})\right) dQ(x,\tau ) \\
&\geq &\frac{1}{2^{p-1}}\int \Lambda _{p}\left( \mathbf{u}_{\tau }(x;\hat{%
\sigma})\right) dQ(x,\tau )-\int \Lambda _{p}\left( -\mathbf{\hat{s}}_{\tau
}(x)\right) dQ(x,\tau ).
\end{eqnarray*}%
From Assumption A\ref{assumption-A3}, the last term is $O_{P}(\left( \log
n\right) ^{p/2})$. Using Assumption A3, we bound the leading integral from
below by%
\begin{equation}
\min_{j\in \mathbb{N}_{J}}r_{n,j}^{p}\left( \int \Lambda _{p}\left( \mathbf{%
\tilde{v}}_{n,\tau }(x)\right) dQ(x,\tau )\left\{ \frac{\int \Lambda
_{p}\left( \mathbf{v}_{n,\tau }(x)\right) dQ(x,\tau )}{\int \Lambda
_{p}\left( \mathbf{\tilde{v}}_{n,\tau }(x)\right) dQ(x,\tau )}-1\right\}
+o_{P}(1)\right) ,  \label{min}
\end{equation}%
where $\mathbf{v}_{n,\tau }(x)\equiv \left[ v_{n,\tau ,j}(x)/\sigma _{n,\tau
,j}(x)\right] _{j\in \mathbb{N}_{J}}$ and $\mathbf{\tilde{v}}_{n,\tau
}(x)\equiv \left[ v_{\tau ,j}(x)/\sigma _{n,\tau ,j}(x)\right] _{j\in 
\mathbb{N}_{J}}$. Since 
\begin{equation*}
\text{liminf}_{n\rightarrow \infty }\int \Lambda _{p}\left( \mathbf{\tilde{v}%
}_{n,\tau }(x)\right) dQ(x,\tau )>0,
\end{equation*}%
we use Assumption C1 and apply the Dominated Convergence Theorem to write (%
\ref{min}) as%
\begin{equation*}
\min_{j\in \mathbb{N}_{J}}r_{n,j}^{p}\int \Lambda _{p}\left( \mathbf{\tilde{v%
}}_{n,\tau }(x)\right) dQ(x,\tau )\left( 1+o_{P}(1)\right) .
\end{equation*}%
Since $\min_{j\in \mathbb{N}_{J}}r_{n,j}\rightarrow \infty $ as $%
n\rightarrow \infty $ and liminf$_{n\rightarrow \infty }\int \Lambda
_{p}\left( \mathbf{\tilde{v}}_{n,\tau }(x)\right) dQ(x,\tau )>0$, we have
for any $M_n \rightarrow \infty$ such that $M_n/{\min_{j \in \mathbb{N}_J} r_{n,j}} \rightarrow 0$, and $M_n/\sqrt{\log n} \rightarrow \infty$,
\begin{equation*}
P\left\{ \frac{1}{2^{p-1}}\int \Lambda _{p}\left( \mathbf{u}_{\tau }(x;\hat{%
\sigma})\right) dQ(x,\tau )>M_n\right\} \rightarrow 1,
\end{equation*}%
as $n\rightarrow \infty $. Also since $\sqrt{\log n}/\min_{j\in \mathbb{N}%
_{J}}r_{n,j}\rightarrow 0$ (Assumption A4(i)), Assumption A\ref%
{assumption-A3} implies that
\begin{equation*}
P\left\{ \hat{\theta}>M_n\right\} \rightarrow 1.
\end{equation*}%
Also, note that by Lemma A2(ii), $h^{-d/2}(c_{\alpha }^{\ast }-a_{n})/\sigma
_{n}=O_{P}(1)$. Hence 
\begin{equation*}
c_{\alpha }^{\ast }=a_{n}+O_{P}(h^{d/2})=O_{P}(1)\text{.}
\end{equation*}%
Given that $c_{\alpha }^{\ast }=O_{P}(1)$ and $\hat{a}^{\ast }=O_{P}(1)$ by
Lemma A1 and Assumption A6(i), we obtain that $P\{\hat{\theta}>c_{\alpha
,\eta }^{\ast }\} \ge P\{\hat{\theta}> M_n\} + o(1) \rightarrow 1$, as $n\rightarrow \infty $.
\end{proof}

\begin{LemmaA}
Suppose that the conditions of Theorem 4 or Theorem 5 hold. Then \textit{as }%
$n\rightarrow \infty ,$ the following holds: for any $c_{n,1},c_{n,2}>0$
such that 
\begin{equation*}
\sqrt{\log n}/c_{n,2}\rightarrow 0,
\end{equation*}%
as $n\rightarrow \infty $. Then%
\begin{equation*}
\inf_{P\in \mathcal{P}_{n}^{0}(\lambda _{n})}P\left\{ \int_{\mathcal{S}%
\backslash B_{n}^{0}(c_{n,1},c_{n,2})}\Lambda _{p}\left( \mathbf{\hat{u}}%
_{\tau }(x)\right) dQ(x,\tau )=0\right\} \rightarrow 1\text{\textit{.}}
\end{equation*}%
Furthermore, we have for any $A\in \mathcal{N}_{J},$%
\begin{equation*}
\inf_{P\in \mathcal{P}_{n}^{0}(\lambda _{n})}P\left\{
\int_{B_{n,A}^{0}(c_{n,1},c_{n,2})}\left\{ \Lambda _{p}\left( \mathbf{\hat{u}%
}_{\tau }(x)\right) -\Lambda _{A,p}\left( \mathbf{\hat{u}}_{\tau }(x)\right)
\right\} dQ(x,\tau )=0\right\} \rightarrow 1\text{.}
\end{equation*}
\end{LemmaA}

\begin{proof}[Proof of Lemma A5]
Consider the first statement. Let $\lambda $ be either $d/2$ or $d/4$. We
write%
\begin{eqnarray*}
&&\int_{\mathcal{S}\backslash B_{n}^{0}(c_{n,1},c_{n,2})}\Lambda _{p}\left( 
\mathbf{\hat{u}}_{\tau }(x)\right) dQ(x,\tau ) \\
&=&\int_{\mathcal{S}\backslash B_{n}^{0}(c_{n,1},c_{n,2})}\Lambda _{p}\left( 
\mathbf{\hat{s}}_{\tau }(x)+\mathbf{u}_{\tau }(x;\hat{\sigma})\right)
dQ(x,\tau ). \\
&=&\int_{\mathcal{S}\backslash B_{n}^{0}(c_{n,1},c_{n,2})}\Lambda _{p}\left( 
\mathbf{\hat{s}}_{\tau }(x)+\mathbf{u}_{\tau }^{0}(x;\hat{\sigma}%
)+h^{\lambda }\delta _{\tau ,\hat{\sigma}}(x)\right) dQ(x,\tau ),
\end{eqnarray*}%
where $\mathbf{u}_{\tau }^{0}(x;\hat{\sigma})\equiv (r_{n,1}v_{n,\tau
,1}^{0}(x)/\hat{\sigma}_{\tau ,1}(x),...,r_{n,J}v_{n,\tau
,J}^{0}(x)/\hat{\sigma}_{\tau ,J}(x))$ and%
\begin{equation}
\delta _{\tau ,\hat{\sigma}}(x)\equiv \left( \frac{\delta _{\tau ,1}(x)}{%
\hat{\sigma}_{\tau ,1}(x)},...,\frac{\delta _{\tau ,J}(x)}{%
\hat{\sigma}_{\tau ,J}(x)}\right) \mathbf{.}  \label{v_sigh_0}
\end{equation}%
Note that $\delta _{\tau ,\hat{\sigma}}(x)$ is bounded with probability
approaching one by Assumption A3. Also note that for each $j\in \mathbb{N}%
_{J}$,%
\begin{align}
\sup_{(x,\tau )\in \mathcal{S}}\left\vert \frac{r_{n,j}\{\hat{v}_{n,\tau
,j}(x)-v_{n,\tau ,j}^{0}(x)\}}{\hat{\sigma}_{\tau ,j}(x)}\right\vert & \leq
\sup_{(x,\tau )\in \mathcal{S}}\left\vert \frac{r_{n,j}\{\hat{v}_{n,\tau
,j}(x)-v_{n,\tau ,j}(x)\}}{\hat{\sigma}_{\tau ,j}(x)}\right\vert +h^{\lambda
}\sup_{(x,\tau )\in \mathcal{S}}\left\vert \frac{\delta _{\tau ,j}(x)}{\hat{%
\sigma}_{\tau ,j}(x)}\right\vert  \label{vrate} \\
& =O_{P}\left( \sqrt{\log n}+h^{\lambda }\right) =O_{P}\left( \sqrt{\log n}%
\right) ,  \notag
\end{align}%
by Assumption A\ref{assumption-A3}. Hence we obtain the desired result,
using the same arguments as in the proof of Lemma 1.

Given that we have (\ref{vrate}), the proof of the second statement can 
proceed in the same way as the proof of the first statement.
\end{proof}

Recall the definitions of $\bar{\Lambda}_{x,\tau }(\mathbf{v})$ in (\ref%
{lambe}). We define for $\mathbf{v}\in \mathbf{R}^{J},$ $\bar{\Lambda}%
_{x,\tau }^{0}(\mathbf{v})$ to be $\bar{\Lambda}_{x,\tau }(\mathbf{v})$
except that $B_{n,A}(c_{n,1},c_{n,2})$ is replaced by $%
B_{n,A}^{0}(c_{n,1},c_{n,2})$. Define for $\lambda \in \{0,d/4,d/2\},$%
\begin{equation}
\hat{\theta}_{\delta }(c_{n,1},c_{n,2};\lambda )\equiv \int \bar{\Lambda}%
_{x,\tau }^{0}\left( \mathbf{\hat{s}}_{\tau }(x)+h^{\lambda }\delta _{\tau
,\sigma }(x)\right) dQ(x,\tau ).  \label{theta_del_e}
\end{equation}%
Let%
\begin{equation*}
a_{n,\delta }^{R}(c_{n,1},c_{n,2};\lambda )\equiv \int \mathbf{E}\left[ \bar{%
\Lambda}_{x,\tau }^{0}\left( \sqrt{nh^{d}}\mathbf{z}_{N,\tau }(x)+h^{\lambda
}\delta _{\tau ,\sigma }(x)\right) \right] dQ(x,\tau )\text{,}
\end{equation*}%
\begin{equation*}
\hat{\theta}_{\delta }^{\ast }(c_{n,1},c_{n,2};\lambda )\equiv \int \bar{%
\Lambda}_{x,\tau }^{0}\left( \mathbf{\hat{s}}_{\tau }^{\ast }(x)+h^{\lambda
}\delta _{\tau ,\sigma }(x)\right) dQ(x,\tau ),
\end{equation*}%
and%
\begin{equation}
a_{n,\delta }^{R\ast }(c_{n,1},c_{n,2};\lambda )\equiv \int \mathbf{E}^{\ast
}\left[ \bar{\Lambda}_{x,\tau }^{0}\left( \sqrt{nh^{d}}\mathbf{z}_{N,\tau
}^{\ast }(x)+h^{\lambda }\delta _{\tau ,\sigma }(x)\right) \right] dQ(x,\tau
).  \label{atilde2}
\end{equation}%
We also define%
\begin{equation*}
a_{n,\delta }(c_{n,1},c_{n,2};\lambda )\equiv \int \mathbf{E}\left[ \bar{%
\Lambda}_{x,\tau }^{0}(\mathbb{W}_{n,\tau ,\tau }^{(1)}(x,0)+h^{\lambda
}\delta _{\tau ,\sigma }(x))\right] dQ(x,\tau ).
\end{equation*}%
When $c_{n,1}=c_{n,2}=c_{n}$, we simply write $a_{n,\delta
}^{R}(c_{n};\lambda ),$ $a_{n,\delta }^{R\ast }(c_{n};\lambda )$, and $%
a_{n,\delta }(c_{n};\lambda )$, instead of writing $a_{n,\delta
}^{R}(c_{n},c_{n};\lambda ),$ $a_{n,\delta }^{R\ast }(c_{n},c_{n};\lambda )$%
, and $a_{n,\delta }(c_{n},c_{n};\lambda )$.

\begin{LemmaA}
\textit{Suppose that the conditions of Assumptions A\ref{assumption-A6}(i)
and B\ref{assumption-B4} hold. Then for each }$P\in \mathcal{P}$\textit{\
such that the local alternatives in (\ref{la}) hold with }$%
b_{n,j}=r_{n,j}h^{-\lambda }$, $j=1,...,J$, \textit{for some }%
$\lambda \in \{0,d/4,d/2\}$, \textit{and for all nonnegative sequences }$%
c_{n,1},c_{n,2},$%
\begin{eqnarray*}
\left\vert a_{n,\delta }^{R}(c_{n,1},c_{n,2};\lambda )-a_{n,\delta
}(c_{n,1},c_{n,2};\lambda )\right\vert &=&o(h^{d/2}),\text{\textit{\ and}} \\
\left\vert a_{n,\delta }^{R\ast }(c_{n,1},c_{n,2};\lambda )-a_{n,\delta
}(c_{n,1},c_{n,2};\lambda )\right\vert &=&o_{P}(h^{d/2}).
\end{eqnarray*}
\end{LemmaA}

\begin{proof}[Proof of Lemma A6]
The result follows immediately from Lemma C12 in Appendix C.
\end{proof}

\begin{LemmaA}
\textit{Suppose that the conditions of Theorem 4 are satisfied. Then for
each }$\lambda \in \{0,d/4,d/2\}$, \textit{for each }$P\in \mathcal{P}%
_{n}^{0}(\lambda _{n})$\textit{\ such that the local alternatives in (\ref%
{la}) hold, }%
\begin{eqnarray*}
&&h^{-d/2}\left( \frac{\bar{\theta}_{n,\delta }(c_{n,U},c_{n,L};\lambda
)-a_{n,\delta }^{R}(c_{n,U},c_{n,L};\lambda )}{\sigma _{n}(c_{n,U},c_{n,L})}%
\right) \overset{d}{\rightarrow }N(0,1)\text{ and} \\
&&h^{-d/2}\left( \frac{\bar{\theta}_{n,\delta }^{\ast
}(c_{n,U},c_{n,L};\lambda )-a_{n,\delta }^{R\ast }(c_{n,U},c_{n,L};\lambda )%
}{\sigma _{n}(c_{n,U},c_{n,L})}\right) \overset{d^{\ast }}{\rightarrow }%
N(0,1),\text{ }\mathcal{P}_{n}^{0}(\lambda _{n})\text{-uniformly.}
\end{eqnarray*}
\end{LemmaA}

\begin{proof}[Proof of Lemma A7]
Note that by the definition of $\mathcal{P}_{n}^{0}(\lambda _{n})$, we have%
\begin{equation*}
\liminf_{n\rightarrow \infty }\inf_{P\in \mathcal{P}_{n}^{0}(\lambda
_{n})}\sigma _{n}^{2}(c_{n,U},c_{n,L})\geq \frac{\eta }{\Phi ^{-1}(1-\alpha )%
}.
\end{equation*}%
Hence we can follow the proof of Lemma A2 to obtain the desired results.%
\textit{\ }
\end{proof}

\begin{proof}[Proof of Theorem \protect\ref{Thm4}]
Using Lemma A5, we find that%
\begin{equation*}
\hat{\theta}=\sum_{A\in \mathcal{N}_{J}}\int_{B_{n,A}^{0}(c_{n,U},c_{n,L})}%
\Lambda _{A,p}\left( \mathbf{\hat{s}}_{\tau }(x)+\mathbf{u}_{\tau }(x;\hat{%
\sigma})\right) dQ(x,\tau )
\end{equation*}%
with probability approaching one. We write the leading sum as 
\begin{equation*}
\sum_{A\in \mathcal{N}_{J}}\int_{B_{n,A}^{0}(0)}\Lambda _{A,p}\left( \mathbf{%
\hat{s}}_{\tau }(x)+\mathbf{u}_{\tau }(x;\hat{\sigma})\right) dQ(x,\tau
)+R_{n},
\end{equation*}%
where 
\begin{equation*}
R_{n}\equiv \sum_{A\in \mathcal{N}_{J}}\int_{B_{n,A}^{0}(c_{n,U},c_{n,L})%
\backslash B_{n,A}^{0}(0)}\Lambda _{A,p}\left( \mathbf{\hat{s}}_{\tau }(x)+%
\mathbf{u}_{\tau }(x;\hat{\sigma})\right) dQ(x,\tau ).
\end{equation*}%
We write $h^{-d/2}R_{n}$ as 
\begin{eqnarray*}
&&h^{-d/2}\sum_{A\in \mathcal{N}_{J}}\int_{B_{n,A}^{0}(c_{n,U},c_{n,L})%
\backslash B_{n,A}^{0}(0)}\Lambda _{A,p}\left( 
\begin{array}{c}
\mathbf{\hat{s}}_{\tau }(x)+\mathbf{u}_{\tau }^{0}(x;\hat{\sigma}) \\ 
+h^{d/2}\delta _{\tau ,\hat{\sigma}}(x)(1+o(1))%
\end{array}%
\right) dQ(x,\tau ) \\
&\leq &h^{-d/2}\sum_{A\in \mathcal{N}_{J}}\int_{B_{n,A}^{0}(c_{n,U},c_{n,L})%
\backslash B_{n,A}^{0}(0)}\Lambda _{A,p}\left( \mathbf{\hat{s}}_{\tau
}(x)+h^{d/2}\delta _{\tau ,\hat{\sigma}}(x)(1+o(1))\right) dQ(x,\tau ),
\end{eqnarray*}%
by Assumption C2. We bound the last sum as 
\begin{equation*}
Ch^{-d/2}\sum_{A\in \mathcal{N}_{J}}\left( \sup_{(x,\tau )\in \mathcal{S}}||%
\mathbf{\hat{s}}_{\tau }(x)||\right) ^{p}Q\left(
B_{n,A}^{0}(c_{n,U},c_{n,L})\backslash B_{n,A}^{0}(0)\right) =O_{P}\left(
h^{-d/2}\left( \log n\right) ^{p/2}\lambda _{n}\right) =o_{P}(1)
\end{equation*}%
using Assumption A\ref{assumption-A3} and the rate condition in (\ref{lamq}%
). We conclude that%
\begin{eqnarray}
h^{-d/2}\hat{\theta} &=&h^{-d/2}\sum_{A\in \mathcal{N}_{J}}%
\int_{B_{n,A}^{0}(0)}\Lambda _{A,p}\left( \mathbf{\hat{s}}_{\tau }(x)+%
\mathbf{u}_{\tau }(x;\hat{\sigma})\right) dQ(x,\tau )+o_{P}(1)  \label{eqs}
\\
&=&h^{-d/2}\sum_{A\in \mathcal{N}_{J}}\int_{B_{n,A}^{0}(0)}\Lambda
_{A,p}\left( \mathbf{\hat{s}}_{\tau }(x)+h^{d/2}\delta _{\tau ,\hat{\sigma}%
}(x)\right) dQ(x,\tau )+o_{P}(1),  \notag
\end{eqnarray}%
where the second equality follows by Assumption C2 and by the definition of $%
B_{n,A}^{0}(0)$.

Fix small $\kappa >0$ and define%
\begin{eqnarray*}
\delta _{\tau ,\sigma ,\kappa ,j}^{L}(x) &\equiv &\left\{ 
\begin{array}{c}
\frac{\delta _{\tau ,j}(x)}{(1+\kappa )\sigma _{n,\tau ,j}(x)}\text{ if }%
\delta _{\tau ,j}(x)\geq 0 \\ 
\frac{\delta _{\tau ,j}(x)}{(1-\kappa )\sigma _{n,\tau ,j}(x)}\text{ if }%
\delta _{\tau ,j}(x)<0%
\end{array}%
\right. \text{ and} \\
\delta _{\tau ,\sigma ,\kappa ,j}^{U}(x) &\equiv &\left\{ 
\begin{array}{c}
\frac{\delta _{\tau ,j}(x)}{(1-\kappa )\sigma _{n,\tau ,j}(x)}\text{ if }%
\delta _{\tau ,j}(x)\geq 0 \\ 
\frac{\delta _{\tau ,j}(x)}{(1+\kappa )\sigma _{n,\tau ,j}(x)}\text{ if }%
\delta _{\tau ,j}(x)<0%
\end{array}%
\right. .
\end{eqnarray*}%
Define $\delta _{\tau ,\sigma ,\kappa }^{L}(x)$ and $\delta _{\tau ,\sigma
,\kappa }^{U}(x)$ to be $\mathbf{R}^{J}$-valued maps whose $j$-th entries
are given by $\delta _{\tau ,\sigma ,\kappa ,j}^{L}(x)$ and $\delta _{\tau
,\sigma ,\kappa ,j}^{U}(x)$ respectively. By construction, Assumptions A3
and C2(ii), we have%
\begin{equation*}
P\left\{ \delta _{\tau ,\sigma ,\kappa }^{L}(x)\leq \delta _{\tau ,\hat{%
\sigma}}(x)\leq \delta _{\tau ,\sigma ,\kappa }^{U}(x)\right\} \rightarrow 1,
\end{equation*}%
as $n\rightarrow \infty $. Therefore, with probability approaching one,%
\begin{eqnarray}
\text{   } \hat{\theta}_{\delta ,L}(0;d/2) &\equiv &\sum_{A\in \mathcal{N}%
_{J}}\int_{B_{n,A}^{0}(0)}\Lambda _{A,p}\left( \mathbf{\hat{s}}_{\tau
}(x)+h^{d/2}\delta _{\tau ,\sigma ,\kappa }^{L}(x)\right) dQ(x,\tau )
\label{three_integrals} \\
&\leq &\sum_{A\in \mathcal{N}_{J}}\int_{B_{n,A}^{0}(0)}\Lambda _{A,p}\left( 
\mathbf{\hat{s}}_{\tau }(x)+h^{d/2}\delta _{\tau ,\hat{\sigma}}(x)\right)
dQ(x,\tau )  \notag \\
&\leq &\sum_{A\in \mathcal{N}_{J}}\int_{B_{n,A}^{0}(0)}\Lambda _{A,p}\left( 
\mathbf{\hat{s}}_{\tau }(x)+h^{d/2}\delta _{\tau ,\sigma ,\kappa
}^{U}(x)\right) dQ(x,\tau )\equiv \hat{\theta}_{\delta ,U}(0;d/2).  \notag
\end{eqnarray}%
We conclude from (\ref{eqs}) that%
\begin{equation}
\hat{\theta}_{\delta ,L}(0;d/2)+o_{P}(h^{d/2})\leq \hat{\theta}\leq \hat{%
\theta}_{\delta ,U}(0;d/2)+o_{P}(h^{d/2}).  \label{cv14}
\end{equation}

As for the bootstrap counterpart, note that since $\delta _{\tau ,j}(x)$ is
bounded and $\sigma _{n,\tau ,j}(x)$ is bounded away from zero uniformly
over $(x,\tau )\in \mathcal{S}$ and $n\geq 1$, and hence 
\begin{equation}
\sup_{(x,\tau )\in \mathcal{S}}\left\vert \frac{1}{h^{-d/2}}\frac{\delta
_{\tau ,j}(x)}{\sigma _{n,\tau ,j}(x)}\right\vert \leq Ch^{d/2}\rightarrow 0,
\label{cv6}
\end{equation}%
as $n\rightarrow \infty $. By (\ref{cv6}), the difference between $%
r_{n,j}v_{n,\tau ,j}(x)/\sigma _{n,\tau ,j}(x)$ and $r_{n,j}v_{n,\tau
,j}^{0}(x)/\sigma _{n,\tau ,j}(x)$ vanishes uniformly over $(x,\tau )\in 
\mathcal{S}$. Therefore, combining this with Lemma A3, we find that%
\begin{equation}
P\left\{ \hat{B}_{n}(\hat{c}_{n})\subset B_{n}^{0}(c_{n,U},c_{n,L})\right\}
\rightarrow 1,  \label{cv5}
\end{equation}%
as $n\rightarrow \infty $.

Now with probability approaching one,%
\begin{eqnarray}
\hat{\theta}^{\ast } &=&\sum_{A\in \mathcal{N}_{J}}\int_{\hat{B}_{A}(\hat{c}%
_{n})}\Lambda _{A,p}\left( \mathbf{\hat{s}}_{\tau }^{\ast }(x)\right)
dQ(x,\tau )  \label{decomp} \\
&=&\sum_{A\in \mathcal{N}_{J}}\int_{B_{n,A}^{0}(0)}\Lambda _{A,p}\left( 
\mathbf{\hat{s}}_{\tau }^{\ast }(x)\right) dQ(x,\tau )  \notag \\
&&+\sum_{A\in \mathcal{N}_{J}}\int_{\hat{B}_{A}(\hat{c}_{n})\backslash
B_{n,A}^{0}(0)}\Lambda _{A,p}\left( \mathbf{\hat{s}}_{\tau }^{\ast
}(x)\right) dQ(x,\tau ).  \notag
\end{eqnarray}%
As for the last sum, it is bounded by%
\begin{equation*}
\sum_{A\in \mathcal{N}_{J}}\int_{B_{n,A}^{0}(c_{n,U},c_{n,L})\backslash
B_{n,A}^{0}(0)}\Lambda _{A,p}\left( \mathbf{\hat{s}}_{\tau }^{\ast
}(x)\right) dQ(x,\tau ),
\end{equation*}%
with probability approaching one by (\ref{cv5}). The above sum multiplied by 
$h^{-d/2}$ is bounded by%
\begin{eqnarray*}
&&h^{-d/2}\left( \sup_{(x,\tau )\in \mathcal{S}}||\mathbf{\hat{s}}_{\tau
}^{\ast }(x)||\right) ^{p}\sum_{A\in \mathcal{N}_{J}}Q\left(
B_{n,A}^{0}(c_{n,U},c_{n,L})\backslash B_{n,A}^{0}(0)\right) \\
&=&O_{P^{\ast }}\left( h^{-d/2}(\log n)^{p/2}\lambda _{n}\right) =o_{P^{\ast
}}(1),\ \mathcal{P}\text{-uniformly,}
\end{eqnarray*}%
by Assumption B2 and the rate condition for $\lambda _{n}$. Thus, we
conclude that%
\begin{equation}
\hat{\theta}^{\ast }=\bar{\theta}^{\ast }(0)+o_{P^{\ast }}(h^{d/2}),\text{ }%
\mathcal{P}_{n}^{0}(\lambda _{n})\text{-uniformly,}  \label{cv15}
\end{equation}%
where%
\begin{equation*}
\bar{\theta}^{\ast }(0)\equiv \sum_{A\in \mathcal{N}_{J}}%
\int_{B_{n,A}^{0}(0)}\Lambda _{A,p}\left( \mathbf{\hat{s}}_{\tau }^{\ast
}(x)\right) dQ(x,\tau ).
\end{equation*}%
Let $\bar{c}_{n}^{\alpha \ast }(0)$ be the $(1-\alpha )$-th quantile of the
bootstrap distribution of $\bar{\theta}^{\ast }(0)$ and let $\gamma
_{n}^{\alpha \ast }(0)$ be the $(1-\alpha )$-th quantile of the bootstrap
distribution of 
\begin{equation}
h^{-d/2}\left( \frac{\bar{\theta}^{\ast }(0)-a_{n}^{R\ast }(0)}{\sigma
_{n}(0)}\right) .  \label{c*tilde0}
\end{equation}%
By the definition of $\mathcal{P}_{n}^{0}(\lambda _{n})$, we have $\sigma
_{n}^{2}(0)>\eta /\Phi ^{-1}(1-\alpha )$. Let $a_{\delta ,U}^{R}(0;d/2)$ and 
$a_{\delta ,L}^{R}(0;d/2)$ be $a_{n,\delta }^{R}(0;d/2)$ except that $\delta
_{\tau ,\sigma }$ is replaced by $\delta _{\tau ,\sigma ,\kappa }^{U}$ and $%
\delta _{\tau ,\sigma ,\kappa }^{L}$ respectively. Also, let $a_{\delta
,U}(0;d/2)$ and $a_{\delta ,L}(0;d/2)$ be $a_{n,\delta }(0;d/2)$ except that 
$\delta _{\tau ,\sigma }$ is replaced by $\delta _{\tau ,\sigma ,\kappa
}^{U} $ and $\delta _{\tau ,\sigma ,\kappa }^{L}$ respectively. We bound $P\{%
\hat{\theta}>c_{\alpha ,\eta }^{\ast }\}$ by%
\begin{eqnarray*}
&&P\left\{ h^{-d/2}\left( \frac{\hat{\theta}_{\delta ,U}(0;d/2)-a_{\delta
,U}^{R}(0;d/2)}{\sigma _{n}(0)}\right) >h^{-d/2}\left( \frac{c_{\alpha
}^{\ast }-a_{\delta ,U}^{R}(0;d/2)}{\sigma _{n}(0)}\right) \right\} +o(1) \\
&=&P\left\{ h^{-d/2}\left( \frac{\hat{\theta}_{\delta ,U}(0;d/2)-a_{\delta
,U}^{R}(0;d/2)}{\sigma _{n}(0)}\right) >h^{-d/2}\left( \frac{\bar{c}%
_{n}^{\alpha \ast }(0)-a_{\delta ,U}^{R}(0;d/2)}{\sigma _{n}(0)}\right)
\right\} +o(1),
\end{eqnarray*}%
where the equality uses (\ref{cv15}). Then we observe that%
\begin{eqnarray*}
\frac{\bar{c}_{n}^{\alpha \ast }(0)-a_{\delta ,U}^{R}(0;d/2)}{\sigma _{n}(0)}
&=&\frac{\bar{c}_{n}^{\alpha \ast }(0)-a_{n}^{R\ast }(0)}{\sigma _{n}(0)}+%
\frac{a_{n}^{R\ast }(0)-a_{\delta ,U}^{R}(0;d/2)}{\sigma _{n}(0)} \\
&=&h^{d/2}\gamma _{n}^{\alpha \ast }(0)+\frac{a_{n}^{R\ast }(0)-a_{\delta
,U}^{R}(0;d/2)}{\sigma _{n}(0)}.
\end{eqnarray*}%
As for the last term, we use Lemmas A1 and A6 to deduce that%
\begin{eqnarray*}
a_{n}^{R\ast }(0)-a_{\delta ,U}^{R}(0;d/2) &=&a_{n}^{R}(0)-a_{\delta
,U}^{R}(0;d/2)+o_{P}(h^{d/2}) \\
&=&a_{n}(0)-a_{\delta ,U}(0;d/2)+o_{P}(h^{d/2}).
\end{eqnarray*}%
As for $a_{n}(0)-a_{\delta ,U}(0;d/2)$, we observe that%
\begin{eqnarray}
&&\sigma _{n}(0)^{-1}h^{-d/2}\left\{ \mathbf{E}\left[ \Lambda _{A,p}\left( 
\mathbb{W}_{n,\tau ,\tau }^{(1)}(x,0)+h^{d/2}\delta _{\tau ,\sigma ,\kappa
}^{U}(x)\right) \right] -\mathbf{E}\left[ \Lambda _{A,p}(\mathbb{W}_{n,\tau
,\tau }^{(1)}(x,0))\right] \right\}  \label{eq55} \\
&=&\sigma _{n}(0)^{-1}h^{-d/2}\left\{ \mathbf{E}\left[ \Lambda _{A,p}\left( 
\mathbb{W}_{n,\tau ,\tau }^{(1)}(x,0)+h^{d/2}\delta _{\tau ,\sigma ,\kappa
}^{U}(x)\right) \right] -\mathbf{E}\left[ \Lambda _{A,p}(\mathbb{W}_{n,\tau
,\tau }^{(1)}(x,0))\right] \right\}  \notag \\
&=&\psi _{n,A,\tau }^{(1)}(\mathbf{0};x)^{\top }\delta _{\tau ,\sigma
,\kappa }^{U}(x)+O\left( h^{d/2}\right) ,  \notag
\end{eqnarray}%
so that%
\begin{eqnarray*}
\frac{h^{-d/2}\left( a_{n}(0)-a_{\delta ,U}(0)\right) }{\sigma _{n}(0)}
&=&-\sum_{A\in \mathcal{N}_{J}}\int \psi _{n,A,\tau }^{(1)}(\mathbf{0}%
;x)^{\top }\delta _{\tau ,\sigma ,\kappa }^{U}(x)dQ(x,\tau )+o(1) \\
&=&-\sum_{A\in \mathcal{N}_{J}}\int \psi _{A,\tau }^{(1)}(\mathbf{0}%
;x)^{\top }\delta _{\tau ,\sigma ,\kappa }^{U}(x)dQ(x,\tau )+o(1),
\end{eqnarray*}%
where the last equality follows by the Dominated Convergence Theorem. On the
other hand, by Lemma A7, we have 
\begin{equation*}
h^{-d/2}\left( \frac{\hat{\theta}_{\delta ,U}(0;d/2)-a_{\delta ,U}^{R}(0;d/2)%
}{\sigma _{n}(0)}\right) \overset{d}{\rightarrow }N(0,1).
\end{equation*}%
Since $\gamma _{n}^{\alpha \ast }(0)=\gamma _{\alpha ,\infty }+o_{P}(1)$ by
Lemma A4, we use this result to deduce that%
\begin{eqnarray*}
&&\lim_{n\rightarrow \infty }P\left\{ h^{-d/2}\left( \frac{\hat{\theta}%
_{\delta ,U}(0;d/2)-a_{\delta ,U}^{R}(0;d/2)}{\sigma _{n}(0)}\right)
>h^{-d/2}\left( \frac{\bar{c}_{n}^{\alpha \ast }(0)-a_{\delta ,U}^{R}(0;d/2)%
}{\sigma _{n}(0)}\right) \right\} \\
&=&1-\Phi \left( z_{1-\alpha }-\sum_{A\in \mathcal{N}_{J}}\int \psi _{A,\tau
}^{(1)}(\mathbf{0};x)^{\top }\delta _{\tau ,\sigma ,\kappa }^{U}(x)dQ(x,\tau
)\right) .
\end{eqnarray*}%
Similarly, we also use (\ref{cv14}) to bound $P\left\{ \hat{\theta}%
>c_{\alpha ,\eta }^{\ast }\right\} $ from below by%
\begin{equation*}
P\left\{ h^{-d/2}\left( \frac{\hat{\theta}_{\delta ,L}(0;d/2)-a_{\delta
,L}^{R}(0;d/2)}{\sigma _{n}(0)}\right) >h^{-d/2}\left( \frac{\bar{c}%
_{n}^{\alpha \ast }(0)-a_{\delta ,L}^{R}(0;d/2)}{\sigma _{n}(0)}\right)
\right\} +o(1),
\end{equation*}%
and using similar arguments as before, we obtain that%
\begin{eqnarray*}
&&\lim_{n\rightarrow \infty }P\left\{ h^{-d/2}\left( \frac{\hat{\theta}%
_{\delta ,L}(0;d/2)-a_{\delta ,L}^{R}(0;d/2)}{\sigma _{n}(0)}\right)
>h^{-d/2}\left( \frac{\bar{c}_{n}^{\alpha \ast }(0)-a_{\delta ,L}^{R}(0;d/2)%
}{\sigma _{n}(0)}\right) \right\} \\
&=&1-\Phi \left( z_{1-\alpha }-\sum_{A\in \mathcal{N}_{J}}\int \psi _{A,\tau
}^{(1)}(\mathbf{0};x)^{\top }\delta _{\tau ,\sigma ,\kappa }^{L}(x)dQ(x,\tau
)\right) .
\end{eqnarray*}%
We conclude from this and (\ref{three_integrals}) that for any small $\kappa
>0$, 
\begin{eqnarray*}
&&1-\Phi \left( z_{1-\alpha }-\sum_{A\in \mathcal{N}_{J}}\int \psi _{A,\tau
}^{(1)}(\mathbf{0};x)^{\top }\delta _{\tau ,\sigma ,\kappa }^{L}(x)dQ(x,\tau
)\right) +o(1) \\
&\leq &P\left\{ \hat{\theta}>c_{\alpha ,\eta }^{\ast }\right\} \leq 1-\Phi
\left( z_{1-\alpha }-\sum_{A\in \mathcal{N}_{J}}\int \psi _{A,\tau }^{(1)}(%
\mathbf{0};x)^{\top }\delta _{\tau ,\sigma ,\kappa }^{U}(x)dQ(x,\tau
)\right) +o(1).
\end{eqnarray*}%
Note that $\psi _{A,\tau }^{(1)}(\mathbf{0};x)^{\top }\delta _{\tau ,\sigma
,\kappa }^{U}(x)$ and $\psi _{A,\tau }^{(1)}(\mathbf{0};x)^{\top }\delta
_{\tau ,\sigma ,\kappa }^{L}(x)$ are bounded maps in $(x,\tau )$ by the
assumption of the theorem, and that 
\begin{equation*}
\lim_{\kappa \rightarrow 0}\delta _{\tau ,\sigma ,\kappa
}^{L}(x)=\lim_{\kappa \rightarrow 0}\delta _{\tau ,\sigma ,\kappa
}^{U}(x)=\delta _{\tau ,\sigma }(x),
\end{equation*}%
for each $(x,\tau )\in \mathcal{S}$. Hence by sending $\kappa \rightarrow 0$
and applying the Dominated Convergence Theorem to both the bounds above, we
obtain the desired result.
\end{proof}

\begin{proof}[Proof of Theorem \protect\ref{Thm5}]
First, observe that Lemma A5 continues to hold. This can be seen by
following the proof of Lemma A5 and noting that (\ref{vrate}) becomes here 
\begin{equation*}
\sup_{(x,\tau )\in \mathcal{S}}\left\vert \frac{r_{n,j}\{\hat{v}_{n,\tau
,j}(x)-v_{n,\tau ,j}^{0}(x)\}}{\hat{\sigma}_{\tau ,j}(x)}\right\vert
=O_{P}\left( \sqrt{\log n}+h^{d/4}\right) =O_{P}\left( \sqrt{\log n}\right) ,
\end{equation*}%
yielding the same convergence rate. The rest of the proof is the same.
Similarly, Lemma A6 continues to hold also under the modified local
alternatives of (\ref{la}) with $b_{n,j}=r_{n,j}h^{-d/4}$. We define 
\begin{equation}
\tilde{\delta}_{\tau ,\sigma }(x)\equiv h^{-d/4}\delta _{\tau ,\sigma }(x).
\label{delta_tilde}
\end{equation}%
We follow the proof of Theorem 4 and take up arguments from (\ref{eq55}).
Observe that%
\begin{eqnarray*}
&&\sigma _{n}(0)^{-1}h^{-d/2}\left\{ \mathbf{E}\left[ \Lambda _{A,p}\left( 
\mathbb{W}_{n,\tau ,\tau }^{(1)}(x,0)+h^{d/2}\tilde{\delta}_{\tau ,\sigma
}(x)\right) \right] -\mathbf{E}\left[ \Lambda _{A,p}(\mathbb{W}_{n,\tau
,\tau }^{(1)}(x,0))\right] \right\} \\
&=&\sigma _{n}(0)^{-1}h^{-d/2}\left\{ \mathbf{E}\left[ \Lambda _{A,p}\left( 
\mathbb{W}_{n,\tau ,\tau }^{(1)}(x,0)+h^{d/2}\tilde{\delta}_{\tau ,\sigma
}(x)\right) \right] -\mathbf{E}\left[ \Lambda _{A,p}(\mathbb{W}_{n,\tau
,\tau }^{(1)}(x,0))\right] \right\} \\
&=&\psi _{n,A,\tau }^{(1)}(\mathbf{0};x)^{\top }\tilde{\delta}_{\tau ,\sigma
}(x)+h^{d/2}\tilde{\delta}_{\tau ,\sigma }(x)^{\top }\psi _{n,A,\tau }^{(2)}(%
\mathbf{0};x)\tilde{\delta}_{\tau ,\sigma }(x)/2.
\end{eqnarray*}%
By the Dominated Convergence Theorem, 
\begin{eqnarray*}
\int \psi _{n,A,\tau }^{(1)}(\mathbf{0};x)^{\top }\tilde{\delta}_{\tau
,\sigma }(x)dQ(x,\tau ) &=&\int \psi _{A,\tau }^{(1)}(\mathbf{0};x)^{\top }%
\tilde{\delta}_{\tau ,\sigma }(x)dQ(x,\tau )+o(1)\text{ and} \\
\int \psi _{n,A,\tau }^{(2)}(\mathbf{0};x)^{\top }\tilde{\delta}_{\tau
,\sigma }(x)dQ(x,\tau ) &=&\int \psi _{A,\tau }^{(2)}(\mathbf{0};x)^{\top }%
\tilde{\delta}_{\tau ,\sigma }(x)dQ(x,\tau )+o(1).
\end{eqnarray*}%
Since $\sum_{A\in \mathcal{N}_{J}}\int \psi _{A,\tau }^{(1)}(\mathbf{0}%
;x)^{\top }\tilde{\delta}_{\tau ,\sigma }(x)dQ(x,\tau )=0$, by the condition
for $\delta _{\tau ,\sigma }(x)$ in the theorem,%
\begin{eqnarray*}
&&\sum_{A\in \mathcal{N}_{J}}\int h^{-d/2}\left\{ 
\begin{array}{c}
\mathbf{E}\left[ \Lambda _{A,p}\left( \mathbb{W}_{n,\tau ,\tau
}^{(1)}(x,0)+h^{d/2}\tilde{\delta}_{\tau ,\sigma }(x)\right) \right] \\ 
-\mathbf{E}\left[ \Lambda _{A,p}(\mathbb{W}_{n,\tau ,\tau }^{(1)}(x,0))%
\right]%
\end{array}%
\right\} dQ(x,\tau ) \\
&=&\frac{1}{2}\sum_{A\in \mathcal{N}_{J}}\int \delta _{\tau ,\sigma
}(x)^{\top }\psi _{A,\tau }^{(2)}(\mathbf{0};x)\delta _{\tau ,\sigma
}(x)dQ(x,\tau )+o(1).
\end{eqnarray*}%
Now we can use the above result by replacing $\delta _{\tau ,\sigma }(x)$ by 
$\delta _{\tau ,\sigma ,\kappa }^{U}(x)$ and $\delta _{\tau ,\sigma ,\kappa
}^{L}(x)$ and follow the proof of Theorem 4 to obtain the desired result.
\end{proof}

\section{Proofs of Auxiliary Results for Lemmas A2(i), Lemma A4(i), and
Theorem 1}

\label{appendix:C}

The eventual result in this appendix is Lemma B9 which is used to show the
asymptotic normality of the location-scale normalized representation of $%
\hat{\theta}$ and its bootstrap version, and to establish its asymptotic
behavior in the degenerate case. For this, we first prepare Lemmas B1-B3. To
obtain uniformity that covers the case of degeneracy, this paper uses a
method of regularization, where the covariance matrix of random quantities
is added by a diagonal matrix of small diagonal elements. The regularized
random quantities having this covariance matrix do not suffer from
degeneracy in the limit, even when the original quantities have covariate
matrix that is degenerate in the limit. Thus, for these regularized
quantities, we can obtain uniform asymptotic theory using an appropriate
Berry-Esseen-type bound. Then, we need to deal with the difference between
the regularized covariance matrix and the original one. Lemma B1 is a simple
result of linear algebra that is used to control this discrepancy.

Lemma B2 has two sub-results from which one can deduce a uniform version of
Levy's continuity theorem. We have not seen any such results in the
literature or monographs, so we provide its full proof. The result has two
functions. First, the result enables one to deduce convergence in
distribution in terms of convergence of cumulative distribution functions
and convergence in distribution in terms of convergence of characteristic
functions in a manner that is uniform over a given collection of
probabilities. The original form of convergence in distribution due to the
Poissonization method in \citeasnoun{GMZ} is convergence of characteristic
functions. Certainly pointwise in $P$, this convergence implies convergence
of cumulative distribution functions, but it is not clear under what
conditions this implication is uniform over a given class of probabilities.
Lemma B2 essentially clarifies this issue.

Lemma B3 is an extension of the de-Poissonization lemma that appeared in %
\citeasnoun{Beirlant/Mason:95}. The proof is based on the proof of their
same result in \citeasnoun{GMZ}, but involves a substantial modification,
because unlike their results, we need a version that holds uniformly over $%
P\in \mathcal{P}$. This de-Poissonization lemma is used to transform the
asymptotic distribution theory for the Poissonized version of the test
statistic into that for the original test statistic.

Lemmas B4-B5 establish some moment bounds for a normalized sum of
independent quantities. This moment bound is later used to control a
Berry-Esseen-type bound, when we approximate those sums by corresponding
centered normal random vectors.

Lemma B6 obtains an approximate version for the scale normalizer $\sigma
_{n} $. The approximate version involves a functional of a Gaussian random
vector, which stems from approximating a normalized sum of independent
random vectors by a Gaussian random vector through using a Berry-Esseen-type
bound. For this result, we use the regularization method that we mentioned
before. Due to the regularization, we are able to cover the degenerate case
eventually.

Lemma B7 is an auxiliary result that is used to establish Lemma B9 in
combination with the de-Poissonization lemma (Lemma B3). And Lemma B8
establishes asymptotic normality of the Poissonized version of the test
statistics. The asymptotic normality for the Poissonized statistic involves
the discretization of the integrals, thereby, reducing the integral to a sum
of 1-dependent random variables, and then applies the Berry-Esseen-type
bound in \citeasnoun{Shergin:93}. Note that by the moment bound in Lemmas
B4-B5 that is uniform over $P\in \mathcal{P}$, we obtain the asymptotic
approximation that is uniform over $P\in \mathcal{P}$. The lemma also
presents a corresponding result for the degenerate case.

Finally, Lemma B9 combines the asymptotic distribution theory for the
Poissonized test statistic in Lemma B7 with the de-Poissonization lemma
(Lemma B3) to obtain the asymptotic distribution theory for the original
test statistic. The result of Lemma B9 is used to establish the asymptotic
normality result in Lemma A7.

The following lemma provides some inequality of matrix algebra.

\begin{LemmaC}
\label{lem-c1} \textit{For any} $J\times J$ \textit{positive semidefinite
symmetric matrix} $\Sigma $ \textit{and any} $\varepsilon >0,$%
\begin{equation*}
\left\Vert \left( \Sigma +\varepsilon I\right) ^{1/2}-\Sigma
^{1/2}\right\Vert \leq \sqrt{J\varepsilon },
\end{equation*}%
\textit{where} $\left\Vert A\right\Vert =\sqrt{tr(AA^{\prime })}$ \textit{%
for any square matrix} $A$.
\end{LemmaC}

\begin{rem}
The main point of Lemma B1 is that the bound $\sqrt{J\varepsilon }$ is
independent of the matrix $\Sigma $. Such a uniform bound is crucially used
for the derivation of asymptotic validity of the test uniform in $P\in 
\mathcal{P}$.
\end{rem}

\begin{proof}[Proof of Lemma B\protect\ref{lem-c1}]
First observe that%
\begin{eqnarray}
&&tr\{\left( \Sigma +\varepsilon I\right) ^{1/2}-\Sigma ^{1/2}\}^{2}
\label{eq} \\
&=&tr\left( 2\Sigma +\varepsilon I\right) -2tr(\left( \Sigma +\varepsilon
I\right) ^{1/2}\Sigma ^{1/2}).  \notag
\end{eqnarray}%
Since $\Sigma \leq \Sigma +\varepsilon I$, we have $\Sigma ^{1/2}\leq \left(
\Sigma +\varepsilon I\right) ^{1/2}.$ For any positive semidefinite matrices 
$A$ and $B,$ $tr(AB)\geq 0$ (see e.g. \citeasnoun{Abadir:Magnus:05}).
Therefore, $tr(\Sigma )\leq tr(\left( \Sigma +\varepsilon I\right)
^{1/2}\Sigma ^{1/2}).$ From (\ref{eq}), we find that 
\begin{eqnarray*}
&&tr\left( 2\Sigma +\varepsilon I\right) -2tr(\left( \Sigma +\varepsilon
I\right) ^{1/2}\Sigma ^{1/2}) \\
&\leq &tr\left( 2\Sigma +\varepsilon I\right) -2tr(\Sigma )=\varepsilon J.
\end{eqnarray*}
\end{proof}

The following lemma can be used to derive a version of Levy's Continuity
Theorem that is uniform in $P\in \mathcal{P}$.

\begin{LemmaC}
\label{lem-c2} \textit{Suppose that }$V_{n}\in \mathbf{R}^{d}$ \textit{is a
sequence of random vectors and} $V\in \mathbf{R}^{d}$ \textit{is a random
vector}. \textit{We assume without loss of generality that }$V_{n}$\textit{\
and }$V$ \textit{live on the same measure space }$(\Omega ,\mathcal{F})$, 
\textit{and $\mathcal{P}$ is a given collection of probabilities on }$%
(\Omega ,\mathcal{F})$. \textit{Furthermore define}%
\begin{eqnarray*}
\varphi _{n}(t) &\equiv &\mathbf{E}\left[ \exp (it^{\top }V_{n})\right] 
\text{, }\varphi (t)\equiv \mathbf{E}\left[ \exp (it^{\top }V)\right] \text{,%
} \\
F_{n}(t) &\equiv &P\left\{ V_{n}\leq t\right\} \text{\textit{, and} }%
F(t)\equiv P\left\{ V\leq t\right\} .
\end{eqnarray*}%
\noindent (i) \textit{Suppose that the distribution }$P\circ V^{-1}$ \textit{%
is uniformly tight in }$\{P\circ V^{-1}:P\in \mathcal{P}\}$. \textit{Then
for any continuous function }$f$\textit{\ on }$\mathbf{R}^{d}$\textit{\
taking values in }$[-1,1]$ \textit{and for any }$\varepsilon \in (0,1],$%
\textit{\ we have}%
\begin{equation*}
\sup_{P\in \mathcal{P}}\left\vert \mathbf{E}f(V_{n})-\mathbf{E}%
f(V)\right\vert \leq \varepsilon ^{-d}C_{d}\sup_{P\in \mathcal{P}}\sup_{t\in 
\mathbf{R}^{d}}\left\vert F_{n}(t)-F(t)\right\vert +4\varepsilon ,
\end{equation*}%
\textit{where }$C_{d}>0$ \textit{is a constant that depends only on} $d$%
\textit{.}

\noindent (ii) \textit{Suppose that }$\sup_{P\in \mathcal{P}}\mathbf{E||}%
V||^{2}<\infty $. \textit{If }%
\begin{equation*}
\sup_{P\in \mathcal{P}}\sup_{u\in \mathbf{R}^{d}}\left\vert \varphi
_{n}(u)-\varphi (u)\right\vert \rightarrow 0\text{\textit{, as }}%
n\rightarrow \infty \text{\textit{,}}
\end{equation*}%
\textit{then for each }$t\in \mathbf{R}^{d},$%
\begin{equation*}
\sup_{P\in \mathcal{P}}\left\vert F_{n}(t)-F(t)\right\vert \rightarrow 0,\ 
\text{\textit{as }}n\rightarrow \infty \text{.}
\end{equation*}%
\textit{\ On the other hand, if for each }$t\in \mathbf{R}^{d},$\textit{\ }%
\begin{equation*}
\sup_{P\in \mathcal{P}}\left\vert F_{n}(t)-F(t)\right\vert \rightarrow 0,\ 
\text{\textit{as }}n\rightarrow \infty ,
\end{equation*}%
\textit{then for each }$u\in \mathbf{R}^{d}$\textit{, }%
\begin{equation*}
\sup_{P\in \mathcal{P}}\left\vert \varphi _{n}(u)-\varphi (u)\right\vert
\rightarrow 0\text{\textit{, as }}n\rightarrow \infty \text{\textit{.}}
\end{equation*}
\end{LemmaC}

\begin{proof}[Proof of Lemma B\protect\ref{lem-c2}]
(i) The proof uses arguments in the proof of Lemma 2.2 of %
\citeasnoun{VanDerVaart:98}. Take a large compact rectangle $B\subset 
\mathbf{R}^{d}$ such that $P\{V\notin B\}<\varepsilon $. Since the
distribution of $V$ is tight uniformly over $P\in \mathcal{P}$, we can take
such $B$ independently of $P\in \mathcal{P}$. Take a partition $B=\cup
_{j=1}^{J_{\varepsilon }}B_{j}$ and points $x_{j}\in B_{j}$ such that $%
J_{\varepsilon }\leq C_{d,1}\varepsilon ^{-d}$, and $|f(x)-f_{\varepsilon
}(x)|<\varepsilon $ for all $x\in B$, where $C_{d,1}>0$ is a constant that
depends only on $d$, and%
\begin{equation*}
f_{\varepsilon }(x)\equiv \sum_{j=1}^{J_{\varepsilon }}f(x_{j})1\{x\in
B_{j}\}.
\end{equation*}%
Thus we have%
\begin{eqnarray*}
\left\vert \mathbf{E}f(V_{n})-\mathbf{E}f(V)\right\vert &\leq &\left\vert 
\mathbf{E}f(V_{n})-\mathbf{E}f_{\varepsilon }(V_{n})\right\vert +\left\vert 
\mathbf{E}f_{\varepsilon }(V_{n})-\mathbf{E}f_{\varepsilon }(V)\right\vert
+\left\vert \mathbf{E}f_{\varepsilon }(V)-\mathbf{E}f(V)\right\vert \\
&\leq &2\varepsilon +P\{V_{n}\notin B\}+P\{V\notin B\}+\left\vert \mathbf{E}%
f_{\varepsilon }(V_{n})-\mathbf{E}f_{\varepsilon }(V)\right\vert \\
&\leq &4\varepsilon +\left\vert P\{V_{n}\notin B\}-P\{V\notin B\}\right\vert
+\left\vert \mathbf{E}f_{\varepsilon }(V_{n})-\mathbf{E}f_{\varepsilon
}(V)\right\vert \\
&=&4\varepsilon +\left\vert P\{V_{n}\in B\}-P\{V\in B\}\right\vert
+\left\vert \mathbf{E}f_{\varepsilon }(V_{n})-\mathbf{E}f_{\varepsilon
}(V)\right\vert .
\end{eqnarray*}%
The second inequality following by $P\{V\notin B\}<\varepsilon $. As for the
last term, we let%
\begin{equation*}
b_{n}\equiv \sup_{P\in \mathcal{P}}\sup_{t\in \mathbf{R}^{d}}\left\vert
F_{n}(t)-F(t)\right\vert ,
\end{equation*}%
and observe that%
\begin{eqnarray*}
\left\vert \mathbf{E}f_{\varepsilon }(V_{n})-\mathbf{E}f_{\varepsilon
}(V)\right\vert &\leq &\sum_{j=1}^{J_{\varepsilon }}\left\vert P\{V_{n}\in
B_{j}\}-P\{V\in B_{j}\}\right\vert |f(x_{j})| \\
&\leq &\sum_{j=1}^{J_{\varepsilon }}\left\vert P\{V_{n}\in B_{j}\}-P\{V\in
B_{j}\}\right\vert \leq C_{d,2}b_{n}J_{\varepsilon },
\end{eqnarray*}%
where $C_{d,2}>0$ is a constant that depends only on $d$. The last
inequality follows because for any rectangle $B_{j}$, we have $|P\{V_{n}\in
B_{j}\}-P\{V\in B_{j}\}|\leq C_{d,2}b_{n}$ for some $C_{d,2}>0.$ We conclude
that%
\begin{equation*}
\left\vert \mathbf{E}f(V_{n})-\mathbf{E}f(V)\right\vert \leq 4\varepsilon
+C_{d,2}\left( C_{d,1}\varepsilon ^{-d}+1\right) b_{n}\leq 4\varepsilon
+C_{d}\varepsilon ^{-d}b_{n},
\end{equation*}%
where $C_{d}=C_{d,2}\{C_{d,1}+1\}$. The last inequality follows because $%
\varepsilon \leq 1$.

\noindent (ii) We show the first statement. We first show that under the
stated condition, the sequence $\{P\circ V_{n}^{-1}\}_{n=1}^{\infty }$ is
uniformly tight over $P\in \mathcal{P}$. That is, for any $%
\varepsilon >0$, we show there exists a compact set $B\subset \mathbf{R}^{d}$
such that for all $n\geq 1,$%
\begin{equation*}
\sup_{P\in \mathcal{P}}P\left\{ V_{n}\in \mathbf{R}^{d}\backslash B\right\}
<\varepsilon \text{.}
\end{equation*}%
For this, we assume $d=1$ without loss of generality, let $P_{n}$ denote the
distribution of $V_{n}$ and consider the following: (using arguments in the
proof of Theorem 3.3.6 of \citeasnoun{Durrett:10})%
\begin{eqnarray*}
P\left\{ |V_{n}|>\frac{2}{u}\right\} &\leq &2\int_{|x|>2/u}\left( 1-\frac{1}{%
|ux|}\right) dP_{n}(x) \\
&\leq &2\int \left( 1-\frac{\sin ux}{ux}\right) dP_{n}(x) \\
&=&\frac{1}{u}\int_{-u}^{u}\left( 1-\varphi _{n}(t)\right) dt.
\end{eqnarray*}%
Define $\bar{e}_{n}\equiv \sup_{P\in \mathcal{P}}\sup_{t\in \mathbf{R}%
}\left\vert \varphi _{n}(t)-\varphi (t)\right\vert $. Using Theorem 3.3.8 of %
\citeasnoun{Durrett:10}, we bound the last term by%
\begin{eqnarray*}
2\bar{e}_{n}+\frac{1}{u}\int_{-u}^{u}\left( 1-\varphi (t)\right) dt &\leq &2%
\bar{e}_{n}+\left\vert \frac{1}{u}\int_{-u}^{u}\left( -it\mathbf{E}V+\frac{%
t^{2}\mathbf{E}V^{2}}{2}\right) dt\right\vert \\
&&+2\left\vert \frac{1}{u}\int_{-u}^{u}t^{2}\mathbf{E}V^{2}dt\right\vert .
\end{eqnarray*}%
The supremum of the right hand side terms over $P\in \mathcal{P}$ vanishes
as we send $n\rightarrow \infty $ and then $u\downarrow 0$, by the
assumption that sup$_{P\in \mathcal{P}}\mathbf{E|}V|^{2}<\infty $. Hence the
sequence $\{P\circ V_{n}^{-1}\}_{n=1}^{\infty }$ is uniformly tight 
over $P\in \mathcal{P}$.

Now, for each $t\in \mathbf{R}^{d}$, there exists a subsequence $\{n^{\prime
}\}\subset \{n\}$ and $\{P_{n^{\prime }}\}\subset \mathcal{P}$ such that%
\begin{equation}
\underset{n\rightarrow \infty }{\text{limsup}}\ \sup_{P\in \mathcal{P}%
}\left\vert F_{n}(t)-F(t)\right\vert =\lim_{n^{\prime }\rightarrow \infty
}\left\vert F_{n^{\prime }}(t;P_{n^{\prime }})-F(t;P_{n^{\prime
}})\right\vert ,  \label{eq5}
\end{equation}%
where%
\begin{equation*}
F_{n}(t;P_{n})=P_{n}\left\{ V_{n}\leq t\right\} \text{ and }%
F(t;P_{n})=P_{n}\left\{ V\leq t\right\} .
\end{equation*}

Since $\{P_{n^{\prime }}\circ V_{n^{\prime }}^{-1}\}_{n^{\prime }=1}^{\infty
}$ is uniformly tight (as shown above), there exists a subsequence $%
\{n_{k}^{\prime }\}\subset \{n^{\prime }\}$ such that 
\begin{equation}
F_{n_{k}^{\prime }}(t;P_{n_{k}^{\prime }})\rightarrow F^{\ast }(t)\text{, as 
}k\rightarrow \infty \text{,}  \label{cv12}
\end{equation}%
for some cdf $F^{\ast }$. Also $\{P_{n^{\prime }}\circ V^{-1}\}_{n^{\prime
}=1}^{\infty }$ is uniformly tight (because $\sup_{P\in \mathcal{P}}\mathbf{E%
}||V||^{2}<\infty $), $\{P_{n_{k}^{\prime }}\circ V^{-1}\}_{k=1}^{\infty }$
is uniformly tight and hence there exists a further subsequence $%
\{n_{k_{j}}^{\prime }\}\subset \{n_{k}^{\prime }\}$ such that 
\begin{equation}
F(t;P_{n_{k_{j}}^{\prime }})\rightarrow F^{\ast \ast }(t)\text{, as }%
j\rightarrow \infty \text{,}  \label{cv25}
\end{equation}%
for some cdf $F^{\ast \ast }$. Since $\{n_{k_{j}}^{\prime }\}\subset
\{n_{k}^{\prime }\}$, we have from (\ref{cv12}),%
\begin{equation}
F_{n_{k_{j}}^{\prime }}(t;P_{n_{k_{j}}^{\prime }})\rightarrow F^{\ast }(t)%
\text{, as }j\rightarrow \infty \text{.}  \label{cv67}
\end{equation}%
By the condition of (ii), we have 
\begin{equation}
\left\vert \varphi _{n_{k_{j}}^{\prime }}(u;P_{n_{k_{j}}^{\prime }})-\varphi
(u;P_{n_{k_{j}}^{\prime }})\right\vert \rightarrow 0\text{, as }j\rightarrow
\infty \text{,}  \label{cv88}
\end{equation}%
where 
\begin{equation*}
\varphi _{n}\left( u;P_{n}\right) =\mathbf{E}_{P_{n}}\left( \exp \left(
iuV_{n}\right) \right) \text{ and }\varphi \left( u;P_{n}\right) =\mathbf{E}%
_{P_{n}}\left( \exp \left( iuV\right) \right) ,
\end{equation*}%
and $\mathbf{E}_{P_{n}}$ represents expectation with respect to the
probability measure $P_{n}$. Furthermore, by (\ref{cv25}) and (\ref{cv67}),
and Levy's Continuity Theorem,%
\begin{equation*}
\lim_{j\rightarrow \infty }\varphi _{n_{k_{j}}^{\prime
}}(u;P_{n_{k_{j}}^{\prime }})\text{ and }\lim_{j\rightarrow \infty }\varphi
(u;P_{n_{k_{j}}^{\prime }})
\end{equation*}%
exist and coincide by (\ref{cv88}). Therefore, for all $t\in \mathbf{R}^{d}$,%
\begin{equation*}
F^{\ast \ast }(t)=F^{\ast }(t).
\end{equation*}%
In other words,%
\begin{equation*}
\lim_{n^{\prime }\rightarrow \infty }\left\vert F_{n^{\prime }}\left(
t;P_{n^{\prime }}\right) -F\left( t;P_{n^{\prime }}\right) \right\vert
=\lim_{n^{\prime }\rightarrow \infty }\left\vert F_{n_{k_{j}}^{\prime
}}\left( t;P_{n_{k_{j}}^{\prime }}\right) -F\left( t;P_{n_{k_{j}}^{\prime
}}\right) \right\vert =0.
\end{equation*}%
Therefore, the first statement of (ii) follows by the last limit applied to (%
\ref{eq5}).

Let us turn to the second statement. Again, we show that $\{P\circ
V_{n}^{-1}\}_{n=1}^{\infty }$ is uniformly tight over $P\in \mathcal{%
P}$. Note that given a large rectangle $B$,%
\begin{equation*}
P\left\{ V_{n}\in \mathbf{R}^{d}\backslash B\right\} \leq \left\vert
P\left\{ V_{n}\in \mathbf{R}^{d}\backslash B\right\} -P\left\{ V\in \mathbf{R%
}^{d}\backslash B\right\} \right\vert +P\left\{ V\in \mathbf{R}%
^{d}\backslash B\right\} .
\end{equation*}%
There exists $N$ such that for all $n\geq N$, the first difference vanishes
as $n\rightarrow \infty $, uniformly in $P\in \mathcal{P}$, by the condition
of the lemma. As for the second term, we bound it by%
\begin{equation*}
P\left\{ V_{j}>a_{j},\ j=1,...,d\right\} \leq \sum_{j=1}^{d}%
\frac{\mathbf{E}V_{j}^{2}}{a_{j}},
\end{equation*}%
where $V_{j}$ is the $j$-th entry of $V$ and $B=\times
_{j=1}^{d}[a_{j},b_{j}]$, $b_{j}<0<a_{j}$. By taking $a_{j}$'s large enough,
we make the last bound arbitrarily small independently of $P\in \mathcal{P}$%
, because sup$_{P\in \mathcal{P}}\mathbf{E}V_{j}^{2}<\infty $ for each $%
j=1,...,d$. Therefore, $\{P\circ V_{n}^{-1}\}_{n=1}^{\infty }$
is uniformly tight over $P\in \mathcal{P}$.

Now, we turn to the proof of the second statement of (ii). For each $u\in 
\mathbf{R}^{d}$, there exists a subsequence $\{n^{\prime }\}\subset \{n\}$
and $\{P_{n^{\prime }}\}\subset \mathcal{P}$ such that 
\begin{equation*}
\underset{n\rightarrow \infty }{\text{limsup}}\ \sup_{P\in \mathcal{P}%
}\left\vert \varphi _{n}(u)-\varphi (u)\right\vert =\lim_{n^{\prime
}\rightarrow \infty }\left\vert \varphi _{n^{\prime }}(u;P_{n^{\prime
}})-\varphi (u;P_{n^{\prime }})\right\vert ,
\end{equation*}%
where $\varphi _{n}(u;P_{n})=\mathbf{E}_{P_{n}}\exp (iu^{\top }V_{n})$ and $%
\varphi (u;P_{n})=\mathbf{E}_{P_{n}}\exp (iu^{\top }V)$. By the condition in
the second statement of (ii), for each $t\in \mathbf{R}^{d}$,%
\begin{equation}
\lim_{n^{\prime }\rightarrow \infty }\left\vert F_{n^{\prime }}\left(
t;P_{n^{\prime }}\right) -F\left( t;P_{n^{\prime }}\right) \right\vert =0%
\text{.}  \label{cv54}
\end{equation}

Since $\{P_{n^{\prime }}\circ V_{n^{\prime }}^{-1}\}_{n^{\prime }=1}^{\infty
}$ is uniformly tight (as shown above), there exists a subsequence $%
\{n_{k}^{\prime }\}\subset \{n^{\prime }\}$ such that $F_{n_{k}^{\prime
}}(t;P_{n_{k}^{\prime }})\rightarrow F^{\ast }(t)$, as $k\rightarrow \infty $%
, and hence by Levy's Continuity Theorem, we have $\varphi _{n_{k}^{\prime
}}(u;P_{n_{k}^{\prime }})\rightarrow \varphi ^{\ast }(u)$, as $k\rightarrow
\infty $. Similarly, we also have $\varphi (u;P_{n_{k}^{\prime
}})\rightarrow \varphi ^{\ast \ast }(u)$, as $k\rightarrow \infty $. By (\ref%
{cv54}), we have $F^{\ast }(t)=F^{\ast \ast }(t)$ and $\varphi ^{\ast
}(u)=\varphi ^{\ast \ast }(u)$. Therefore,%
\begin{equation*}
\lim_{n^{\prime }\rightarrow \infty }\left\vert \varphi _{n^{\prime }}\left(
u;P_{n^{\prime }}\right) -\varphi \left( u;P_{n^{\prime }}\right)
\right\vert =\lim_{n^{\prime }\rightarrow \infty }\left\vert \varphi
_{n_{k_{j}}^{\prime }}\left( u;P_{n_{k_{j}}^{\prime }}\right) -\varphi
\left( u;P_{n_{k_{j}}^{\prime }}\right) \right\vert =0.
\end{equation*}%
Thus we arrive at the desired result.
\end{proof}

The following lemma offers a version of the de-Poissonization lemma of %
\citeasnoun{Beirlant/Mason:95} (see Theorem 2.1 on page 5). In contrast to
the result of \citeasnoun{Beirlant/Mason:95}, the version here is uniform in 
$P\in \mathcal{P}$.

\begin{LemmaC}
\label{lem-c3} \textit{Let} $N_{1,n}(\alpha )$ \textit{and} $N_{2,n}(\alpha
) $ \textit{be independent Poisson random variables with} $N_{1,n}(\alpha )$ 
\textit{being Poisson} $(n(1-\alpha ))$ \textit{and} $N_{2,n}(\alpha )$ 
\textit{being Poisson} $(n\alpha )$, \textit{where} $\alpha \in (0,1)$. 
\textit{Denote} $N_{n}(\alpha )=N_{1,n}(\alpha )+N_{2,n}(\alpha )$ \textit{%
and set}%
\begin{equation*}
U_{n}(\alpha )=\frac{N_{1,n}(\alpha )-n(1-\alpha )}{\sqrt{n}}\text{ \textit{%
and} }V_{n}(\alpha )=\frac{N_{2,n}(\alpha )-n\alpha }{\sqrt{n}}.
\end{equation*}%
\textit{Let }$\{S_{n}\}_{n=1}^{\infty }$ \textit{be a sequence of random
variables and} $\mathcal{P}$ \textit{be a given set of probabilities} $P$ 
\textit{on a measure space on which }$(S_{n},U_{n}(\alpha _{P}),V_{n}(\alpha
_{P}))$ \textit{lives, where} $\alpha _{P}\in (0,1)$ \textit{is a quantity
that may depend on} $P\in \mathcal{P}$ \textit{and for some} $\varepsilon >0$%
, 
\begin{equation}
\varepsilon \leq \inf_{P\in \mathcal{P}}\alpha _{P}\leq \sup_{P\in \mathcal{P%
}}\alpha _{P}\leq 1-\varepsilon .  \label{bds}
\end{equation}

\textit{Furthermore, assume that for each} $n\geq 1$, \textit{the random
vector} $(S_{n},U_{n}(\alpha _{P}))$\textit{\ is independent of }$%
V_{n}(\alpha _{P})$ \textit{with respect to each} $P\in \mathcal{P}$. 
\textit{Let for }$t_{1},t_{2}\in \mathbf{R}^{2}$,%
\begin{equation*}
b_{n,P}(t_{1},t_{2};\sigma _{P})\equiv \left\vert P\left\{ S_{n}\leq
t_{1},U_{n}(\alpha _{P})\leq t_{2}\right\} -P\{\sigma _{P}\mathbb{Z}_{1}\leq
t_{1},\sqrt{1-\alpha _{P}}\mathbb{Z}_{2}\leq t_{2}\}\right\vert ,
\end{equation*}%
\textit{where }$\mathbb{Z}_{1}$ \textit{and} $\mathbb{Z}_{2}$\textit{\ are
independent standard normal random variables and }$\sigma _{P}^{2}>0$\textit{%
\ for each }$P\in \mathcal{P}$. \textit{(Note that }$\inf_{P\in \mathcal{P}%
}\sigma _{P}^{2}$ \textit{is allowed to be zero.})\newline
(i) \textit{As} $n\rightarrow \infty ,$%
\begin{eqnarray*}
&&\sup_{P\in \mathcal{P}}\sup_{t\in \mathbf{R}}\left\vert \mathbf{E}[\exp
(itS_{n})|N_{n}(\alpha _{P})=n]-\exp \left( -\frac{\sigma _{P}^{2}t^{2}}{2}%
\right) \right\vert \\
&\leq &2\varepsilon +\left( 4C_{d}\sup_{P\in \mathcal{P}}a_{n,P}(\varepsilon
)\right) \sqrt{\frac{2\pi }{\varepsilon }},
\end{eqnarray*}%
\textit{where }$a_{n,P}(\varepsilon )\equiv \varepsilon
^{-d}b_{n,P}+\varepsilon ,b_{n,P}\equiv \sup_{t_{1},t_{2}\in \mathbf{R}%
}b_{n,P}(t_{1},t_{2};\sigma _{P})$,\textit{\ and }$\varepsilon $\textit{\ is
the constant in }(\ref{bds}).\newline
(ii) \textit{Suppose further that for all }$t_{1},t_{2}\in \mathbf{R}$, 
\textit{as} $n\rightarrow \infty ,$%
\begin{equation*}
\sup_{P\in \mathcal{P}}b_{n,P}(t_{1},t_{2};0)\rightarrow 0.
\end{equation*}%
\textit{Then, for all }$t\in \mathbf{R}$, \textit{we have as }$n\rightarrow
\infty ,$%
\begin{equation*}
\sup_{P\in \mathcal{P}}\left\vert \mathbf{E}[\exp (itS_{n})|N_{n}(\alpha
_{P})=n]-1\right\vert \rightarrow 0.
\end{equation*}
\end{LemmaC}

\begin{rem}
While the proof of Lemma B3 follows that of Lemma 2.4 of \citeasnoun{GMZ},
it is worth noting that in contrast to Lemma 2.4 of \citeasnoun{GMZ} or
Theorem 2.1 of \citeasnoun{Beirlant/Mason:95}, Lemma B3 gives an explicit
bound for the difference between the conditional characteristic function of $%
S_{n}$ given $N_{n}(\alpha _{P})=n$ and the characteristic function of $%
N(0,\sigma _{P}^{2})$. Under the stated conditions, (in particular (\ref{bds}%
)), the explicit bound is shown to depend on $P\in \mathcal{P}$ only through 
$b_{n,P} $. Thus in order to obtain a bound uniform in $P\in \mathcal{P}$,
it suffices to control $\alpha _{P}$ and $b_{n,P}$ uniformly in $P\in 
\mathcal{P}$.
\end{rem}

\begin{proof}[Proof of Lemma B\protect\ref{lem-c3}]
(i) Let $\phi _{n,P}(t,u)=\mathbf{E}[\exp (itS_{n}+iuU_{n}(\alpha _{P}))]$
and%
\begin{equation*}
\phi _{P}(t,u)=\exp (-(\sigma _{P}^{2}t^{2}+(1-\alpha _{P})u^{2})/2).
\end{equation*}%
By the condition of the lemma and Lemma B2(i), we have for any $\varepsilon
>0,$%
\begin{eqnarray}
\left\vert \phi _{n,P}(t,u)-\phi _{P}(t,u)\right\vert &\leq &(\varepsilon
^{-d}C_{d}b_{n,P}+4\varepsilon )  \label{cv44} \\
&\leq &4\varepsilon ^{-d}C_{d}b_{n,P}+4\varepsilon
=4C_{d}a_{n,P}(\varepsilon )\text{.}  \notag
\end{eqnarray}%
Note that $a_{n,P}(\varepsilon )$ depends on $P\in \mathcal{P}$ only through 
$b_{n,P}$.

Following the proof of Lemma 2.4 of \citeasnoun{GMZ}, we have%
\begin{eqnarray*}
\psi _{n,P}(t) &=&\mathbf{E}[\exp (itS_{n})|N_{n}(\alpha _{P})=n] \\
&=&\frac{1}{\sqrt{2\pi }}\left( 1+o(1)\right) \int_{-\pi \sqrt{n}}^{\pi 
\sqrt{n}}\phi _{n,P}(t,v)\mathbf{E}\left[ \exp (ivV_{n}(\alpha _{P}))\right]
dv,
\end{eqnarray*}%
uniformly over $P\in \mathcal{P}$. Note that the equality comes after
applying Sterling's formula to $2\pi P\{N_{n}(\alpha _{P})=n\}$ and change
of variables from $u$ to $v/\sqrt{n}$. (See the proof of Lemma 2.4 of %
\citeasnoun{GMZ}.) The distribution of $N_{n}(\alpha _{P}),$ being Poisson $%
(n)$, does not depend on the particular choice of $\alpha _{P}\in (0,1),$
and hence the $o(1)$ term is $o(1)$ uniformly over $t\in \mathbf{R}$ and
over $P\in \mathcal{P}$. We follow the proof of Theorem 3 of %
\citeasnoun[p.517]{Feller:66} to observe that there exists $n_{0}>0$ such
that uniformly over $\alpha \in \lbrack \varepsilon ,1-\varepsilon ],$%
\begin{equation*}
\left\{ \int_{-\pi \sqrt{n}}^{\pi \sqrt{n}}\left\vert \mathbf{E}\exp
(ivV_{n}(\alpha ))-\exp (-\alpha v^{2}/2)\right\vert dv+\int_{|v|>\pi \sqrt{n%
}}\exp \left( -\alpha v^{2}/2\right) dv\right\} <\varepsilon ,
\end{equation*}%
for all $n>n_{0}$. Note that the distribution of $V_{n}(\alpha _{P})$
depends on $P\in \mathcal{P}$ only through $\alpha _{P}\in \lbrack
\varepsilon ,1-\varepsilon ]$ and $\varepsilon $ does not depend on $P$.
Since there exists $n_{1}$ such that for all $n>n_{1}$, 
\begin{equation*}
\sup_{P\in \mathcal{P}}\int_{|v|>\pi \sqrt{n}}\exp \left( -\alpha
_{P}v^{2}/2\right) dv<\varepsilon ,
\end{equation*}%
the previous inequality implies that for all $n>\max \{n_{0},n_{1}\}$,%
\begin{eqnarray}
&&\sup_{P\in \mathcal{P}}\int_{-\pi \sqrt{n}}^{\pi \sqrt{n}}\left\vert \phi
_{n,P}(t,u)\left( \mathbf{E}\exp (iuV_{n}(\alpha _{P}))-\exp (-\alpha
_{P}u^{2}/2)\right) \right\vert du  \label{cv23} \\
&\leq &\sup_{P\in \mathcal{P}}\int_{-\pi \sqrt{n}}^{\pi \sqrt{n}}\left(
\sup_{P\in \mathcal{P}}|\phi _{n,P}(t,u)|\right) |\mathbf{E}\exp
(iuV_{n}(\alpha _{P}))-\exp (-\alpha _{P}u^{2}/2)|du  \notag \\
&\leq &\sup_{P\in \mathcal{P}}\int_{-\pi \sqrt{n}}^{\pi \sqrt{n}}|\mathbf{E}%
\exp (iuV_{n}(\alpha _{P}))-\exp (-\alpha _{P}u^{2}/2)|du\leq \varepsilon . 
\notag
\end{eqnarray}%
By (\ref{cv44}) and (\ref{cv23}),%
\begin{eqnarray*}
&&\sup_{P\in \mathcal{P}}\left\vert \int_{-\pi \sqrt{n}}^{\pi \sqrt{n}}\phi
_{n,P}(t,u)\mathbf{E}\left[ \exp (iuV_{n}(\alpha _{P}))\right] du-\int_{-\pi 
\sqrt{n}}^{\pi \sqrt{n}}\phi _{P}(t,u)\exp \left( -\alpha _{P}u^{2}/2\right)
du\right\vert \\
&\leq &\sup_{P\in \mathcal{P}}\sup_{\alpha \in \lbrack \varepsilon
,1-\varepsilon ]}\int_{-\pi \sqrt{n}}^{\pi \sqrt{n}}\left\vert \phi
_{n,P}(t,u)\left( \mathbf{E}\exp (iuV_{n}(\alpha ))-\exp (-\alpha
u^{2}/2)\right) \right\vert du \\
&&+\int_{-\pi \sqrt{n}}^{\pi \sqrt{n}}\sup_{P\in \mathcal{P}}\sup_{\alpha
\in \lbrack \varepsilon ,1-\varepsilon ]}\left\vert \phi _{n,P}(t,u)-\phi
_{P}(t,u)\right\vert \exp (-\alpha u^{2}/2)du \\
&\leq &\varepsilon +\left( 4C_{d}\sup_{P\in \mathcal{P}}a_{n,P}(\varepsilon
)\right) \sup_{\alpha \in \lbrack \varepsilon ,1-\varepsilon ]}\int_{-\pi 
\sqrt{n}}^{\pi \sqrt{n}}\exp (-\alpha u^{2}/2)du \\
&\leq &\varepsilon +\left( 4C_{d}\sup_{P\in \mathcal{P}}a_{n,P}(\varepsilon
)\right) \sup_{\alpha \in \lbrack \varepsilon ,1-\varepsilon ]}\sqrt{\frac{%
2\pi }{\alpha }}=\varepsilon +\left( 4C_{d}\sup_{P\in \mathcal{P}%
}a_{n,P}(\varepsilon )\right) \sqrt{\frac{2\pi }{\varepsilon }}
\end{eqnarray*}%
as $n\rightarrow \infty $. Since 
\begin{equation*}
\exp \left( -\frac{\sigma _{P}^{2}t^{2}}{2}\right) =\frac{1}{\sqrt{2\pi }}%
\int_{-\infty }^{\infty }\phi _{P}(t,u)\exp \left( -\frac{\alpha _{P}u^{2}}{2%
}\right) du,
\end{equation*}%
and from some large $n$ that does not depend on $P\in \mathcal{P}$,%
\begin{eqnarray*}
&&\left\vert \int_{-\infty }^{\infty }\phi _{P}(t,u)\exp \left( -\frac{%
\alpha _{P}u^{2}}{2}\right) du-\int_{-\pi \sqrt{n}}^{\pi \sqrt{n}}\phi
_{P}(t,u)\exp \left( -\frac{\alpha _{P}u^{2}}{2}\right) du\right\vert \\
&=&\exp \left( -\frac{\sigma _{P}^{2}t^{2}}{2}\right) \left\vert
\int_{-\infty }^{\infty }\exp \left( -\frac{u^{2}}{2}\right) du-\int_{-\pi 
\sqrt{n}}^{\pi \sqrt{n}}\exp \left( -\frac{u^{2}}{2}\right) du\right\vert
<\varepsilon ,
\end{eqnarray*}%
we conclude that for each $t\in \mathbf{R}$,%
\begin{equation*}
\left\vert \psi _{n,P}(t)-\exp \left( -\frac{\sigma _{P}^{2}t^{2}}{2}\right)
\right\vert \leq 2\varepsilon +\left( 4C_{d}\sup_{P\in \mathcal{P}%
}a_{n,P}(\varepsilon )\right) \sqrt{\frac{2\pi }{\varepsilon }},
\end{equation*}%
as $n\rightarrow \infty $. Since the right hand side does not depend on $%
t\in \mathbf{R}$ and $P\in \mathcal{P}$, we obtain the desired result.%
\newline
(ii) By the condition of the lemma and Lemma B2(ii), we have for any $t,u\in 
\mathbf{R},$%
\begin{equation*}
\sup_{P\in \mathcal{P}}\left\vert \phi _{n,P}(t,u)-\phi _{P}(0,u)\right\vert
\rightarrow 0\text{,}
\end{equation*}%
as $n\rightarrow \infty $. The rest of the proof is similar to that of (i).
We omit the details.
\end{proof}

Define for $x\in \mathcal{X}$, $\tau _{1},\tau _{2}\in \mathcal{T}$, and $%
j,k\in \mathbb{N}_{J},$%
\begin{equation*}
k_{n,\tau ,j,m}(x)\equiv \frac{1}{h^{d}}\mathbf{E}\left[ \left\vert \beta
_{n,x,\tau ,j}\left( Y_{ij},\frac{X_{i}-x}{h}\right) \right\vert ^{m}\right].
\end{equation*}

\begin{LemmaC}
\label{lem-c4} \textit{Suppose that Assumption A\ref{assumption-A6}(i) holds.%
} \textit{Then for all }$m\in \lbrack 2,M]$, \textit{(with }$M>2$ \textit{%
being the constant that appears in Assumption A\ref{assumption-A6}(i)}), 
\textit{there exists }$C_{1}\in (0,\infty )$ \textit{that does not depend on}
$n$ \textit{such that for each }$j\in \mathbb{N}_{J}$,%
\begin{equation*}
\sup_{\tau \in \mathcal{T},x\in \mathcal{S}_{\tau }(\varepsilon )}\sup_{P\in 
\mathcal{P}}k_{n,\tau ,j,m}(x)\leq C_{1}.
\end{equation*}
\end{LemmaC}

\begin{proof}[Proof of Lemma B\protect\ref{lem-c4}]
The proof can proceed by using Assumption A\ref{assumption-A6}(i) and
following the proof of Lemma 4 of \citeasnoun{LSW}.
\end{proof}

Let $N$ be a Poisson random variable with mean $n$ and independent of $%
(Y_{i}^{\top },X_{i}^{\top })_{i=1}^{\infty }$. Also, let $\beta _{n,x,\tau
}(Y_{i},(X_{i}-x)/h)$ be the $J$-dimensional vector whose $j$-th entry is
equal to $\beta _{n,x,\tau ,j}(Y_{ij},(X_{i}-x)/h)$. We define%
\begin{eqnarray*}
\mathbf{z}_{N,\tau }(x) &\equiv &\frac{1}{nh^{d}}\sum_{i=1}^{N}\beta
_{n,x,\tau }\left( Y_{i},\frac{X_{i}-x}{h}\right) -\frac{1}{h^{d}}\mathbf{E}%
\beta _{n,x,\tau }\left( Y_{i},\frac{X_{i}-x}{h}\right) \text{ and} \\
\mathbf{z}_{n,\tau }(x) &\equiv &\frac{1}{nh^{d}}\sum_{i=1}^{n}\beta
_{n,x,\tau }\left( Y_{i},\frac{X_{i}-x}{h}\right) -\frac{1}{h^{d}}\mathbf{E}%
\beta _{n,x,\tau }\left( Y_{i},\frac{X_{i}-x}{h}\right) .
\end{eqnarray*}%
Let $N_{1}$ be a Poisson random variable with mean 1, independent of $%
(Y_{i}^{\top },X_{i}^{\top })_{i=1}^{\infty }$. Define%
\begin{eqnarray*}
q_{n,\tau }(x) &\equiv &\frac{1}{\sqrt{h^{d}}}\sum_{1\leq i\leq
N_{1}}\left\{ \beta _{n,x,\tau }\left( Y_{i},\frac{X_{i}-x}{h}\right) -%
\mathbf{E}\beta _{n,x,\tau }\left( Y_{i},\frac{X_{i}-x}{h}\right) \right\} 
\text{ and} \\
\bar{q}_{n,\tau }(x) &\equiv &\frac{1}{\sqrt{h^{d}}}\left\{ \beta _{n,x,\tau
}\left( Y_{i},\frac{X_{i}-x}{h}\right) -\mathbf{E}\beta _{n,x,\tau }\left(
Y_{i},\frac{X_{i}-x}{h}\right) \right\} .
\end{eqnarray*}

\begin{LemmaC}
\label{lem-c5} \textit{Suppose that Assumption A\ref{assumption-A6}(i)
holds. Then for any }$m\in \lbrack 2,M]$\textit{\ (with }$M>2$\textit{\
being the constant in Assumption A6(i))}%
\begin{eqnarray}
\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\mathbf{E}\left[
||q_{n,\tau }(x)||^{m}\right] &\leq &\bar{C}_{1}h^{d(1-(m/2))}\text{\textit{%
\ and}}  \label{bd52} \\
\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\mathbf{E}\left[ ||%
\bar{q}_{n,\tau }(x)||^{m}\right] &\leq &\bar{C}_{2}h^{d(1-(m/2))},  \notag
\end{eqnarray}%
\textit{where }$\bar{C}_{1},\bar{C}_{2}>0$ \textit{are constants that depend
only on} $m.$

\textit{If furthermore, } $\limsup_{n\rightarrow \infty
}n^{-(m/2)+1}h^{d(1-(m/2))}<C$ \textit{for some constant} $C>0$, \textit{then%
} 
\begin{eqnarray}
\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\mathbf{E}\left[
||n^{1/2}h^{d/2}\mathbf{z}_{N,\tau }(x)||^{m}\right] &\leq &\left( \frac{15m%
}{\log m}\right) ^{m}\max \left\{ \bar{C}_{1},2\bar{C}_{1}C\right\} \text{%
\textit{\ and}}  \label{ineqs} \\
\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\mathbf{E}\left[
||n^{1/2}h^{d/2}\mathbf{z}_{n,\tau }(x)||^{m}\right] &\leq &\left( \frac{15m%
}{\log m}\right) ^{m}\max \left\{ \bar{C}_{2},2\bar{C}_{2}C\right\} ,  \notag
\end{eqnarray}%
\textit{where} $\bar{C}_{1},\bar{C}_{2}>0$\textit{\ are the constants that
appear in (\ref{bd52}).}
\end{LemmaC}

\begin{proof}[Proof of Lemma B\protect\ref{lem-c5}]
Let $q_{n,\tau ,j}(x)$ be the $j$-th entry of $q_{n,\tau }(x)$. For the
first statement of the lemma, it suffices to observe that for some positive
constants $C_{1}$ and $\bar{C},$%
\begin{equation}
\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\mathbf{E}\left[
|q_{n,\tau ,j}(x)|^{m}\right] \leq \frac{C_{1}h^{d}k_{n,\tau ,j,m}}{h^{dm/2}}%
\leq \bar{C}h^{d(1-(m/2))},  \label{bdd}
\end{equation}%
where the first inequality uses the definition of $k_{n,\tau ,j,m}$, and the
last inequality uses Lemma B4 and the fact that $m\in \lbrack 2,M]$. The
second statement in (\ref{bd52}) follows similarly.

We consider the statements in (\ref{ineqs}). We consider the first
inequality in (\ref{ineqs}). Let $z_{N,\tau ,j}(x)$ be the $j$-th entry of $%
\mathbf{z}_{N,\tau }(x)$. Then using Rosenthal's inequality (e.g. (2.3) of %
\citeasnoun{GMZ}), we find that%
\begin{eqnarray*}
&&\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\mathbf{E}[|\sqrt{%
nh^{d}}z_{N,\tau ,j}(x)|^{m}] \\
&\leq &\left( \frac{15m}{\log m}\right) ^{m}\sup_{(x,\tau )\in \mathcal{S}%
}\sup_{P\in \mathcal{P}}\max \left\{ \left( \mathbf{E}q_{n,\tau
,j}^{2}(x)\right) ^{m/2},n^{-m/2+1}\mathbf{E}|q_{n,\tau ,j}(x)|^{m}\right\} .
\end{eqnarray*}%
Since $\mathbf{E}q_{n,\tau ,j}^{2}(x)\leq (\mathbf{E}|q_{n,\tau
,j}(x)|^{m})^{2/m}$, by (\ref{bdd}), the last term is bounded by%
\begin{eqnarray*}
&&\left( \frac{15m}{\log m}\right) ^{m}\max \left\{ \bar{C},\bar{C}%
n^{-(m/2)+1}h^{d(1-(m/2))}\right\} \\
&\leq &\left( \frac{15m}{\log m}\right) ^{m}\max \left\{ \bar{C},2\bar{C}%
C\right\} ,
\end{eqnarray*}%
from some large $n$ by the condition limsup$_{n\rightarrow \infty
}n^{-(m/2)+1}h^{d(1-(m/2))}<C$.

As for the second inequality in (\ref{ineqs}), for some $C>0$, we use the
second inequality in (\ref{bd52}) and use Rosenthal's inequality in the same
way as before, to obtain the inequality.
\end{proof}

The following lemma offers a characterization of the scale normalizer of our
test statistic. For $A,A^{\prime }\subset \mathbb{N}_{J}$, define $\mathbf{%
\zeta }_{n,\tau }(x)\equiv \sqrt{nh^{d}}\mathbf{z}_{N,\tau }(x)$,%
\begin{eqnarray}
C_{n,\tau ,\tau ^{\prime },A,A^{\prime }}^{R}(x,x^{\prime }) &\equiv
&h^{-d}Cov\left( \Lambda _{A,p}\left( \mathbf{\zeta }_{n,\tau }(x)\right)
,\Lambda _{A^{\prime },p}\left( \mathbf{\zeta }_{n,\tau ^{\prime
}}(x^{\prime })\right) \right) \text{, and}  \label{C} \\
C_{n,\tau ,\tau ^{\prime },A,A^{\prime }}(x,u) &\equiv &Cov\left( \Lambda
_{A,p}\left( \mathbb{W}_{n,\tau ,\tau ^{\prime }}^{(1)}(x,u)\right) ,\Lambda
_{A^{\prime },p}\left( \mathbb{W}_{n,\tau ,\tau ^{\prime
}}^{(2)}(x,u)\right) \right) ,  \notag
\end{eqnarray}%
where we recall that $[\mathbb{W}_{n,\tau _{1},\tau _{2}}^{(1)}(x,u)^{\top },%
\mathbb{W}_{n,\tau _{1},\tau _{2}}^{(2)}(x,u)^{\top }]^{\top }$ is a mean
zero $\mathbf{R}^{2J}$-valued Gaussian random vector whose covariance matrix
is given by (\ref{cov}).

Then for Borel sets $B,B^{\prime }\subset \mathcal{S}$ and $A,A^{\prime
}\subset \mathbb{N}_{J},$ let%
\begin{equation*}
\sigma _{n,A,A^{\prime }}^{R}(B,B^{\prime })\equiv \int_{B^{\prime
}}\int_{B}C_{n,\tau ,\tau ^{\prime },A,A^{\prime }}^{R}(x,x^{\prime
})dQ(x,\tau )dQ(x^{\prime },\tau ^{\prime })
\end{equation*}%
and%
\begin{equation}
\sigma _{n,A,A^{\prime }}(B,B^{\prime })\equiv \int_{\mathcal{T}}\int_{%
\mathcal{T}}\int_{B_{\tau }\cap B_{\tau ^{\prime }}^{\prime }}\int_{\mathcal{%
U}}C_{n,\tau ,\tau ^{\prime },A,A^{\prime }}(x,u)dudxd\tau d\tau ^{\prime },
\label{s2}
\end{equation}%
where $B_{\tau }\equiv \{x\in \mathcal{X}:(x,\tau )\in B\}$ and $B_{\tau
^{\prime }}^{\prime }\equiv \{x\in \mathcal{X}:(x,\tau ^{\prime })\in
B^{\prime }\}$.

The lemma below shows that $\sigma _{n,A,A^{\prime }}^{R}(B,B^{\prime })$
and $\sigma _{n,A,A^{\prime }}(B,B^{\prime })$ are asymptotically equivalent
uniformly in $P\in \mathcal{P}$. We introduce some notation. Recall the
definition of $\Sigma _{n,\tau _{1},\tau _{2}}(x,u)$, which is found below (%
\ref{rho}). Define for $\bar{\varepsilon}>0,$%
\begin{equation*}
\tilde{\Sigma}_{n,\tau _{1},\tau _{2},\bar{\varepsilon}}(x,u)\equiv \left[ 
\begin{array}{c}
\Sigma _{n,\tau _{1},\tau _{1}}(x,0)+\bar{\varepsilon}I_{J} \\ 
\Sigma _{n,\tau _{1},\tau _{2}}(x,u)%
\end{array}%
\begin{array}{c}
\Sigma _{n,\tau _{1},\tau _{2}}(x,u) \\ 
\Sigma _{n,\tau _{2},\tau _{2}}(x+uh,0)+\bar{\varepsilon}I_{J}%
\end{array}%
\right] ,
\end{equation*}%
where $I_{J}$ is the $J$ dimensional identity matrix. Certainly $\tilde{%
\Sigma}_{n,\tau _{1},\tau _{2},\bar{\varepsilon}}(x,u)$ is positive
definite. We define%
\begin{equation*}
\mathbf{\xi }_{N,\tau _{1},\tau _{2}}(x,u;\eta _{1},\eta _{2})\equiv \sqrt{%
nh^{d}}\tilde{\Sigma}_{n,\tau _{1},\tau _{2},\bar{\varepsilon}}^{-1/2}(x,u)%
\left[ 
\begin{array}{c}
\mathbf{z}_{N,\tau _{1}}(x;\eta _{1}) \\ 
\mathbf{z}_{N,\tau _{2}}(x+uh;\eta _{2})%
\end{array}%
\right] ,
\end{equation*}%
where $\eta _{1}\in \mathbf{R}^{J}$ and $\eta _{2}\in \mathbf{R}^{J}$ are
random vectors that are independent, and independent of $(Y_{i}^{\top
},X_{i}^{\top })_{i=1}^{\infty }$, each following $N(0,\bar{\varepsilon}%
I_{J})$, and $\mathbf{z}_{N,\tau }(x;\eta _{1})\equiv \mathbf{z}_{N,\tau
}(x)+\eta _{1}/\sqrt{nh^{d}}.\mathbf{\ }$We are prepared to state the lemma.

\begin{LemmaC}
\label{lem-c6} \textit{Suppose that\ Assumption A\ref{assumption-A6}(i)
holds and that }$nh^{d}\rightarrow \infty ,$\textit{\ as }$n\rightarrow
\infty $, \textit{and }%
\begin{equation*}
\underset{n\rightarrow \infty }{\text{limsup}}\ n^{-(m/2)+1}h^{d(1-(m/2))}<C,
\end{equation*}%
\textit{for some constant }$C>0$\textit{\ and some }$m\in \lbrack 2(p+1),M]$.

\textit{Then for any sequences of Borel sets} $B_{n},$ $B_{n}^{\prime
}\subset \mathcal{S}$ \textit{and for any} $A,A^{\prime }\subset \mathbb{N}%
_{J},$%
\begin{equation*}
\sigma _{n,A,A^{\prime }}^{R}(B_{n},B_{n}^{\prime })=\sigma _{n,A,A^{\prime
}}(B_{n},B_{n}^{\prime })+o(1),
\end{equation*}%
\textit{where }$o(1)$ \textit{vanishes uniformly in} $P\in \mathcal{P}$%
\textit{\ as }$n\rightarrow \infty $.
\end{LemmaC}

\begin{rem}
The main innovative element of Lemma B\ref{lem-c6} is that the result does
not require that $\sigma _{n,A,A^{\prime }}(B_{n},B_{n}^{\prime })$ be
positive for each finite $n$ or positive in the limit. Hence the result can
be applied to the case where the scale normalizer $\sigma _{n,A,A^{\prime
}}^{R}(B_{n},B_{n}^{\prime })$ is degenerate (either in finite samples or
asymptotically).
\end{rem}

\begin{proof}[Proof of Lemma B\protect\ref{lem-c6}]
Define $B_{n,\tau }\equiv \{x\in \mathcal{X}:(x,\tau )\in B_{n}\}$, $w_{\tau
,B_{n}}(x)\equiv 1_{B_{n,\tau }}(x).$ For a given $\bar{\varepsilon}>0,$ let 
\begin{eqnarray*}
g_{1n,\tau _{1},\tau _{2},\bar{\varepsilon}}(x,u) &\equiv &h^{-d}Cov(\Lambda
_{A,p}(\sqrt{nh^{d}}\mathbf{z}_{N,\tau _{1}}(x;\eta _{1})),\Lambda
_{A^{\prime },p}(\sqrt{nh^{d}}\mathbf{z}_{N,\tau _{2}}(x+uh;\eta _{2}))), \\
g_{2n,\tau _{1},\tau _{2},\bar{\varepsilon}}(x,u) &\equiv &Cov(\Lambda
_{A,p}(\mathbb{Z}_{n,\tau _{1},\tau _{2},\bar{\varepsilon}}(x)),\Lambda
_{A^{\prime },p}(\mathbb{Z}_{n,\tau _{1},\tau _{2},\bar{\varepsilon}%
}(x+uh))),
\end{eqnarray*}%
and $\left( \mathbb{Z}_{n,\tau _{1},\tau _{2},\bar{\varepsilon}}^{\top }(x),%
\mathbb{Z}_{n,\tau _{1},\tau _{2},\bar{\varepsilon}}^{\top }(v)\right)
^{\top }$ is a centered normal $\mathbf{R}^{2J}$-valued random vector with
the same covariance matrix as that of $[\sqrt{nh^{d}}\mathbf{z}_{N,\tau
_{1}}^{\top }(x;\eta _{1}),\sqrt{nh^{d}}\mathbf{z}_{N,\tau _{2}}^{\top
}(v;\eta _{2})]^{\top }$. Then we define%
\begin{equation*}
\sigma _{n,A,A^{\prime },\bar{\varepsilon}}^{R}(B_{n},B_{n}^{\prime })\equiv
\int_{\mathcal{T}}\int_{\mathcal{T}}\int_{B_{n,\tau _{1}}}\int_{\mathcal{U}%
}g_{1n,\tau _{1},\tau _{2},\bar{\varepsilon}}(x,u)w_{\tau
_{1},B_{n}}(x)w_{\tau _{2},B_{n}^{\prime }}(x+uh)dudxd\tau _{1}d\tau _{2},
\end{equation*}%
and%
\begin{equation*}
\sigma _{n,A,A^{\prime },\bar{\varepsilon}}(B_{n},B_{n}^{\prime })\equiv
\int_{\mathcal{T}}\int_{\mathcal{T}}\int_{B_{n,\tau _{1}}\cap B_{n,\tau
_{2}}^{\prime }}\int_{\mathcal{U}}C_{n,\tau _{1},\tau _{2},A,A^{\prime },%
\bar{\varepsilon}}(x,u)dudxd\tau _{1}d\tau _{2},
\end{equation*}%
where%
\begin{equation}
C_{n,\tau _{1},\tau _{2},A,A^{\prime },\bar{\varepsilon}}(x,u)\equiv
Cov\left( \Lambda _{A,p}(\mathbb{W}_{n,\tau _{1},\tau _{2},\bar{\varepsilon}%
}^{(1)}(x,u)),\Lambda _{A^{\prime },p}(\mathbb{W}_{n,\tau _{1},\tau _{2},%
\bar{\varepsilon}}^{(2)}(x,u))\right) ,  \label{Ce}
\end{equation}%
and, with $\mathbb{Z}\sim N(0,I_{2J})$, 
\begin{equation}
\left[ 
\begin{array}{c}
\mathbb{W}_{n,\tau _{1},\tau _{2},\bar{\varepsilon}}^{(1)}(x,u) \\ 
\mathbb{W}_{n,\tau _{1},\tau _{2},\bar{\varepsilon}}^{(2)}(x,u)%
\end{array}%
\right] \equiv \tilde{\Sigma}_{n,\tau _{1},\tau _{2},\bar{\varepsilon}%
}^{1/2}(x,u)\mathbb{Z}.  \label{Ws}
\end{equation}%
Thus, $\sigma _{n,A,A^{\prime },\bar{\varepsilon}}^{R}(B_{n},B_{n}^{\prime
}) $ and $\sigma _{n,A,A^{\prime },\bar{\varepsilon}}(B_{n},B_{n}^{\prime })$
are \textquotedblleft regularized\textquotedblright\ versions of $\sigma
_{n,A,A^{\prime }}^{R}(B_{n},B_{n}^{\prime })$ and $\sigma _{n,A,A^{\prime
}}(B_{n},B_{n}^{\prime })$. We also define%
\begin{equation*}
\tau _{n,A,A^{\prime },\bar{\varepsilon}}(B_{n},B_{n}^{\prime })\equiv \int_{%
\mathcal{T}}\int_{\mathcal{T}}\int_{B_{n,\tau _{1}}}\int_{\mathcal{U}%
}g_{2n,\tau _{1},\tau _{2},\bar{\varepsilon}}(x,u)w_{\tau
_{1},B_{n}}(x)w_{\tau _{2},B_{n}^{\prime }}(x+uh)dudxd\tau _{1}d\tau _{2}.
\end{equation*}%
Then it suffices for the lemma to show the following two statements. \newline
\textbf{Step 1: }As $n\rightarrow \infty ,$%
\begin{eqnarray*}
\sup_{P\in \mathcal{P}}\left\vert \sigma _{n,A,A^{\prime },\bar{\varepsilon}%
}^{R}(B_{n},B_{n}^{\prime })-\tau _{n,A,A^{\prime },\bar{\varepsilon}%
}(B_{n},B_{n}^{\prime })\right\vert &\rightarrow &0,\text{ and} \\
\sup_{P\in \mathcal{P}}\left\vert \tau _{n,A,A^{\prime },\bar{\varepsilon}%
}(B_{n},B_{n}^{\prime })-\sigma _{n,A,A^{\prime },\bar{\varepsilon}%
}(B_{n},B_{n}^{\prime })\right\vert &\rightarrow &0.
\end{eqnarray*}%
\newline
\textbf{Step 2: }For some $C>0$ that does not depend on $\bar{\varepsilon}\ $%
or $n,$ 
\begin{eqnarray*}
\sup_{P\in \mathcal{P}}|\sigma _{n,A,A^{\prime },\bar{\varepsilon}%
}^{R}(B_{n},B_{n}^{\prime })-\sigma _{n,A,A^{\prime
}}^{R}(B_{n},B_{n}^{\prime })| &\leq &C\sqrt{\bar{\varepsilon}},\text{ and}
\\
\sup_{P\in \mathcal{P}}\left\vert \sigma _{n,A,A^{\prime },\bar{\varepsilon}%
}(B_{n},B_{n}^{\prime })-\sigma _{n,A,A^{\prime }}(B_{n},B_{n}^{\prime
})\right\vert &\leq &C\sqrt{\bar{\varepsilon}}.
\end{eqnarray*}

Then the desired result follows by sending $n\rightarrow \infty $ and then $%
\bar{\varepsilon}\downarrow 0$, while chaining Steps 1 and 2. \newline
\textbf{Proof of Step 1:} We first focus on the first statement. For any
vector $\mathbf{v}=[\mathbf{v}_{1}^{\top },\mathbf{v}_{2}^{\top }]^{\top
}\in \mathbf{R}^{2J}$, we define%
\begin{eqnarray*}
\tilde{\Lambda}_{A,p,1}\left( \mathbf{v}\right) &\equiv &\Lambda
_{A,p}\left( \left[ \tilde{\Sigma}_{n,\tau _{1},\tau _{2},\bar{\varepsilon}%
}^{1/2}(x,u)\mathbf{v}\right] _{1}\right) , \\
\tilde{\Lambda}_{A^{\prime },p,2}\left( \mathbf{v}\right) &\equiv &\Lambda
_{A^{\prime },p}\left( \left[ \tilde{\Sigma}_{n,\tau _{1},\tau _{2},\bar{%
\varepsilon}}^{1/2}(x,u)\mathbf{v}\right] _{2}\right) ,
\end{eqnarray*}%
and%
\begin{equation}
C_{n,p}(\mathbf{v})\equiv \tilde{\Lambda}_{A,p,1}\left( \mathbf{v}\right) 
\tilde{\Lambda}_{A^{\prime },p,2}\left( \mathbf{v}\right) ,  \label{lambda_t}
\end{equation}%
where $[a]_{1}$ of a vector $a\in \mathbf{R}^{2J}$ indicates the vector of
the first $J$ entries of $a$, and $[a]_{2}$ the vector of the remaining $J$
entries of $a.$ By Theorem 9 of \citeasnoun[p. 208]{Magnus:Neudecker:01},%
\begin{eqnarray}
\lambda _{\min }\left( \tilde{\Sigma}_{n,\tau _{1},\tau _{2},\bar{\varepsilon%
}}(x,u)\right) &\geq &\lambda _{\min }\left( \left[ 
\begin{array}{c}
\Sigma _{n,\tau _{1},\tau _{2}}(x,0) \\ 
\Sigma _{n,\tau _{1},\tau _{2}}^{\top }(x,u)%
\end{array}%
\begin{array}{c}
\Sigma _{n,\tau _{1},\tau _{2}}(x,u) \\ 
\Sigma _{n,\tau _{2},\tau _{2}}(x+uh,0)%
\end{array}%
\right] \right)  \label{lb2} \\
&&+\lambda _{\min }\left( \left[ 
\begin{array}{c}
\bar{\varepsilon}I_{J} \\ 
0%
\end{array}%
\begin{array}{c}
0 \\ 
\bar{\varepsilon}I_{J}%
\end{array}%
\right] \right)  \notag \\
&\geq &\lambda _{\min }\left( \left[ 
\begin{array}{c}
\bar{\varepsilon}I_{J} \\ 
0%
\end{array}%
\begin{array}{c}
0 \\ 
\bar{\varepsilon}I_{J}%
\end{array}%
\right] \right) =\bar{\varepsilon}.  \notag
\end{eqnarray}%
Let $q_{n,\tau ,j}(x;\eta _{1j})\equiv p_{n,\tau ,j}(x)+\eta _{1j},$ where 
\begin{equation*}
p_{n,\tau ,j}(x)\equiv \frac{1}{\sqrt{h^{d}}}\sum_{1\leq i\leq N_{1}}\left\{
\beta _{n,x,\tau ,j}\left( Y_{ij},\frac{X_{i}-x}{h}\right) -\mathbf{E}\left[
\beta _{n,x,\tau ,j}\left( Y_{ij},\frac{X_{i}-x}{h}\right) \right] \right\} ,
\end{equation*}%
$\eta _{1j}$ is the $j$-th entry of $\eta _{1}$, and $N_{1}$ is a Poisson
random variable with mean $1$ and $((\eta _{1j})_{j\in \mathbb{N}%
_{J}},N_{1}) $ is independent of $\{(Y_{i}^{\top },X_{i}^{\top
})\}_{i=1}^{\infty }.$ Let $p_{n,\tau }(x)$ be the column vector of entries $%
p_{n,\tau ,j}(x)$ with $j$ running in the set $\mathbb{N}_{J}$. Let $%
[p_{n,\tau _{1}}^{(i)}(x),p_{n,\tau _{2}}^{(i)}(x+uh)]$ be i.i.d. copies of $%
[p_{n,\tau _{1}}(x),p_{n,\tau _{2}}(x+uh)]$ and $\eta _{1}^{(i)}$ and $\eta
_{2}^{(i)}$ be also i.i.d. copies of $\eta _{1}$ and $\eta _{2}$. Define 
\begin{equation*}
q_{n,\tau ,1}^{(i)}(x)\equiv p_{n,\tau }^{(i)}(x)+\eta _{1}^{(i)}\text{ and }%
q_{n,\tau ,2}^{(i)}(x+uh)\equiv p_{n,\tau }^{(i)}(x+uh)+\eta _{2}^{(i)}.
\end{equation*}%
Note that%
\begin{equation*}
\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\left[ 
\begin{array}{c}
q_{n,\tau _{1},1}^{(i)}(x) \\ 
q_{n,\tau _{2},2}^{(i)}(x+uh)%
\end{array}%
\right] =\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\left[ 
\begin{array}{c}
p_{n,\tau _{1}}^{(i)}(x) \\ 
p_{n,\tau _{2}}^{(i)}(x+uh)%
\end{array}%
\right] +\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\left[ 
\begin{array}{c}
\eta _{1}^{(i)} \\ 
\eta _{2}^{(i)}%
\end{array}%
\right] .
\end{equation*}%
The last sum has the same distribution as $[\eta _{1}^{\top },\eta
_{2}^{\top }]^{\top }$ and the leading sum on the right-hand side has the
same distribution as that of $[\mathbf{z}_{N,\tau _{1}}^{\top }(x),\mathbf{z}%
_{N,\tau _{2}}^{\top }(x+uh)]^{\top }$. Therefore, we conclude that%
\begin{equation*}
\mathbf{\xi }_{N,\tau _{1},\tau _{2}}(x,u;\eta _{1},\eta _{2})\overset{d}{=}%
\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\tilde{W}_{n,\tau _{1},\tau _{2}}^{(i)}(x,u),
\end{equation*}%
where%
\begin{equation*}
\tilde{W}_{n,\tau _{1},\tau _{2}}^{(i)}(x,u)\equiv \tilde{\Sigma}_{n,\tau
_{1},\tau _{2},\bar{\varepsilon}}^{-1/2}(x,u)\left[ 
\begin{array}{c}
q_{n,\tau _{1},1}^{(i)}(x) \\ 
q_{n,\tau _{2},2}^{(i)}(x+uh)%
\end{array}%
\right] .
\end{equation*}%
Now we invoke the Berry-Esseen-type bound of 
\citeasnoun[Theorem
1]{Sweeting:77} to prove Step 1. By Lemma B5, we deduce that%
\begin{equation}
\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\mathbf{E}||q_{n,\tau
,1}^{(i)}(x)||^{3}\leq Ch^{-d/2},  \label{bd35}
\end{equation}%
for some $C>0$. Also, recall the definition of $\rho _{n,\tau _{1},\tau
_{1},j,j}(x,0)$ in (\ref{rho}) and note that%
\begin{eqnarray}
&&\sup_{\tau \in \mathcal{T}}\sup_{(x,u)\in \mathcal{S}_{\tau }(\varepsilon
)\times \mathcal{U}}\sup_{P\in \mathcal{P}}tr\left( \tilde{\Sigma}_{n,\tau
_{1},\tau _{2},\bar{\varepsilon}}(x,u)\right)  \label{trace} \\
&\leq &\sup_{\tau \in \mathcal{T},x\in \mathcal{S}_{\tau }(\varepsilon
)}\sup_{P\in \mathcal{P}}\sum_{j\in J}\left( \rho _{n,\tau _{1},\tau
_{1},j,j}(x,0)+\rho _{n,\tau _{2},\tau _{2},j,j}(x,0)+2\bar{\varepsilon}%
\right) \leq C,  \notag
\end{eqnarray}%
for some $C>0$ that depends only on $J$ and $\bar{\varepsilon}$ by Lemma B4.
Observe that by the definition of $C_{n,p}$ in (\ref{lambda_t}), and (\ref%
{trace}),%
\begin{equation*}
\sup_{\mathbf{v}\in \mathbf{R}^{2J}}\frac{\left\vert C_{n,p}(\mathbf{v}%
)-C_{n,p}(0)\right\vert }{1+||\mathbf{v}||^{2p+2}\min \left\{ ||\mathbf{v}%
||,1\right\} }\leq C.
\end{equation*}%
We find that for each $u\in \mathcal{U},$ $||\tilde{W}_{n,\tau _{1},\tau
_{2}}^{(i)}(x,u)||^{2}$ is equal to%
\begin{eqnarray}
&&tr\left( \tilde{\Sigma}_{n,\tau _{1},\tau _{2},\bar{\varepsilon}%
}^{-1/2}(x,u)\left[ 
\begin{array}{c}
q_{n,\tau _{1},1}^{(i)}(x) \\ 
q_{n,\tau _{1},2}^{(i)}(x+uh)%
\end{array}%
\right] \left[ 
\begin{array}{c}
q_{n,\tau _{2},1}^{(i)}(x) \\ 
q_{n,\tau _{2},2}^{(i)}(x+uh)%
\end{array}%
\right] ^{\top }\tilde{\Sigma}_{n,\tau _{1},\tau _{2},\bar{\varepsilon}%
}^{-1/2}(x,u)\right)  \label{der5} \\
&\leq &\lambda _{\max }\left( \tilde{\Sigma}_{n,\tau _{1},\tau _{2},\bar{%
\varepsilon}}^{-1}(x,u)\right) tr\left( \left[ 
\begin{array}{c}
q_{n,\tau _{1},1}^{(i)}(x) \\ 
q_{n,\tau _{1},2}^{(i)}(x+uh)%
\end{array}%
\right] \left[ 
\begin{array}{c}
q_{n,\tau _{2},1}^{(i)}(x) \\ 
q_{n,\tau _{2},2}^{(i)}(x+uh)%
\end{array}%
\right] ^{\top }\right) .  \notag
\end{eqnarray}%
Therefore, $\mathbf{E}||\tilde{W}_{n,\tau _{1},\tau _{2}}^{(i)}(x,u)||^{3}$
is bounded by%
\begin{equation*}
\lambda _{\max }^{3/2}\left( \tilde{\Sigma}_{n,\tau _{1},\tau _{2},\bar{%
\varepsilon}}^{-1}(x,u)\right) \mathbf{E}\left\Vert \left[ 
\begin{array}{c}
q_{n,\tau _{1},1}^{(i)}(x) \\ 
q_{n,\tau _{2},2}^{(i)}(x+uh)%
\end{array}%
\right] \right\Vert ^{3}.
\end{equation*}%
From (\ref{lb2}),%
\begin{equation*}
\lambda _{\max }^{3/2}(\tilde{\Sigma}_{n,\tau _{1},\tau _{2},\bar{\varepsilon%
}}^{-1}(x,u))=\lambda _{\min }^{-3/2}(\tilde{\Sigma}_{n,\tau _{1},\tau _{2},%
\bar{\varepsilon}}(x,u))\leq \bar{\varepsilon}^{-3/2}.
\end{equation*}%
Therefore, we conclude that%
\begin{eqnarray*}
&&\sup_{\tau \in \mathcal{T}}\sup_{(x,u)\in \mathcal{S}_{\tau }(\varepsilon
)\times \mathcal{U}}\sup_{P\in \mathcal{P}}\mathbf{E}||\tilde{W}_{n,\tau
_{1},\tau _{2}}^{(i)}(x,u)||^{3} \\
&\leq &C_{1}\bar{\varepsilon}^{-3/2}\cdot \sup_{\tau \in \mathcal{T},x\in 
\mathcal{S}_{\tau }(\varepsilon )}\sup_{P\in \mathcal{P}}\mathbf{E}%
||q_{n,\tau _{1},1}^{(i)}(x)||^{3} \\
&&+C_{1}\bar{\varepsilon}^{-3/2}\cdot \sup_{\tau \in \mathcal{T}%
}\sup_{(x,u)\in \mathcal{S}_{\tau }(\varepsilon )\times \mathcal{U}%
}\sup_{P\in \mathcal{P}}\mathbf{E}||q_{n,\tau _{2},2}^{(i)}(x+uh)||^{3}\leq
C_{2}\bar{\varepsilon}^{-3/2}/\sqrt{h^{d}},
\end{eqnarray*}%
where $C_{1}>0$ and $C_{2}>0$ are constants depending only on $J$, and the
last bound follows by (\ref{bd35}). Therefore, by Theorem 1 of %
\citeasnoun{Sweeting:77}, we find that with $\bar{\varepsilon}>0$ fixed and $%
n\rightarrow \infty $,%
\begin{eqnarray}
&&\sup_{\tau \in \mathcal{T}}\sup_{(x,u)\in \mathcal{S}_{\tau }(\varepsilon
)\times \mathcal{U}}\sup_{P\in \mathcal{P}}\left\vert \mathbf{E}%
C_{n,p}\left( \frac{1}{\sqrt{n}}\sum_{i=1}^{n}\tilde{W}_{n,\tau _{1},\tau
_{2}}^{(i)}(x,u)\right) -\mathbf{E}C_{n,p}\left( \mathbb{\tilde{Z}}_{n,\tau
_{1},\tau _{2}}(x,u)\right) \right\vert  \label{cv4} \\
&=&O\left( n^{-1/2}h^{-d/2}\right) =o(1),  \notag
\end{eqnarray}%
where $\mathbb{\tilde{Z}}_{n,\tau _{1},\tau _{2}}(x,u)=[\mathbb{Z}_{n,\tau
_{1},\tau _{2},\bar{\varepsilon}}(x)^{\top },\mathbb{Z}_{n,\tau _{1},\tau
_{2},\bar{\varepsilon}}(x+uh)^{\top }]^{\top }.$

Using similar arguments, we also deduce that for $j=1,2,$ and $A\subset 
\mathbb{N}_{J}$,%
\begin{equation*}
\sup_{\tau \in \mathcal{T}}\sup_{(x,u)\in \mathcal{S}_{\tau }(\varepsilon
)\times \mathcal{U}}\sup_{P\in \mathcal{P}}\left\vert \mathbf{E}\tilde{%
\Lambda}_{A,p,j}\left( \frac{1}{\sqrt{n}}\sum_{i=1}^{n}\tilde{W}_{n,\tau
_{1},\tau _{2}}^{(i)}(x,u)\right) -\mathbf{E}\tilde{\Lambda}_{A,p,j}\left( 
\mathbb{\tilde{Z}}_{n,\tau _{1},\tau _{2}}(x,u)\right) \right\vert =o(1).
\end{equation*}

For some $C>0,$%
\begin{eqnarray*}
&&\sup_{\tau \in \mathcal{T}}\sup_{(x,u)\in \mathcal{S}_{\tau }(\varepsilon
)\times \mathcal{U}}\sup_{P\in \mathcal{P}}Cov\left( \Lambda _{p}(\mathbb{Z}%
_{n,\tau _{1},\tau _{2},\bar{\varepsilon}}(x)),\Lambda _{p}(\mathbb{Z}%
_{n,\tau _{1},\tau _{2},\bar{\varepsilon}}(x+uh))\right) \\
&\leq &\sup_{\tau \in \mathcal{T}}\sup_{(x,u)\in \mathcal{S}_{\tau
}(\varepsilon )\times \mathcal{U}}\sup_{P\in \mathcal{P}}\sqrt{\mathbf{E}%
\left\Vert \mathbb{Z}_{n,\tau _{1},\tau _{2},\bar{\varepsilon}%
}(x)\right\Vert ^{2p}}\sqrt{\mathbf{E}\left\Vert \mathbb{Z}_{n,\tau
_{1},\tau _{2},\bar{\varepsilon}}(x+uh)\right\Vert ^{2p}}<C.
\end{eqnarray*}%
The last inequality follows because $\mathbb{Z}_{n,\tau _{1},\tau _{2},\bar{%
\varepsilon}}(x)$ and $\mathbb{Z}_{n,\tau _{1},\tau _{2},\bar{\varepsilon}%
}(x+uh)$ are centered normal random vectors with a covariance matrix that
has a finite Euclidean norm by Lemma B4. Hence we apply the Dominated
Convergence Theorem to deduce the first statement of Step 1 from (\ref{cv4}).%

We turn to the second statement of Step 1. The statement immediately follows
because for each $u\in \mathcal{U},$ the covariance matrix of $\tilde{\Sigma}%
_{n,\tau _{1},\tau _{2},\bar{\varepsilon}}^{-1/2}(x,u)\mathbf{\xi }_{n,\tau
_{1},\tau _{2},\bar{\varepsilon}}(x,u)$ is equal to the covariance matrix of 
$[\mathbb{W}_{n,\tau _{1},\tau _{2},\bar{\varepsilon}}^{(1)\top }(x,u),%
\mathbb{W}_{n,\tau _{1},\tau _{2},\bar{\varepsilon}}^{(2)\top }(x,u)]^{\top
}and$%
\begin{equation*}
\left\vert w_{\tau _{1},B_{n}}(x)w_{\tau _{2},B_{n}^{\prime }}(x+uh)-w_{\tau
_{1},B_{n}}(x)w_{\tau _{2},B_{n}^{\prime }}(x)\right\vert \rightarrow 0,
\end{equation*}%
as $n\rightarrow \infty $, for each $u\in \mathcal{U}$, and for almost every 
$x\in \mathcal{X}$ (with respect to Lebesgue measure.)\newline
\textbf{Proof of Step 2: }We consider the first statement. First, we write%
\begin{eqnarray}
&&\left\vert \left( \sigma _{n,A,A^{\prime },\bar{\varepsilon}%
}^{R}(B_{n},B_{n}^{\prime })\right) ^{2}-\left( \sigma _{n,A,A^{\prime
}}^{R}(B_{n},B_{n}^{\prime })\right) ^{2}\right\vert  \label{ineq44} \\
&\leq &\int_{\mathcal{T}}\int_{\mathcal{T}}\int_{B_{n}}\int_{\mathcal{U}%
}\left\vert \Delta _{n,\tau _{1},\tau _{2},1}^{\eta }(x,u)\right\vert
w_{\tau _{1},B_{n}}(x)w_{\tau _{2},B_{n}^{\prime }}(x+uh)dudxd\tau _{1}d\tau
_{2}  \notag \\
&&+\int_{\mathcal{T}}\int_{\mathcal{T}}\int_{B_{n}}\int_{\mathcal{U}%
}\left\vert \Delta _{n,\tau _{1},\tau _{2},2}^{\eta }(x,u)\right\vert
w_{\tau _{1},B_{n}}(x)w_{\tau _{2},B_{n}^{\prime }}(x+uh)dudxd\tau _{1}d\tau
_{2},  \notag
\end{eqnarray}%
where%
\begin{eqnarray*}
\Delta _{n,\tau _{1},\tau _{2},1}^{\eta }(x,u) &=&\mathbf{E}\Lambda _{A,p}(%
\sqrt{nh^{d}}\mathbf{z}_{N,\tau _{1}}(x))\mathbf{E}\Lambda _{A^{\prime },p}(%
\sqrt{nh^{d}}\mathbf{z}_{N,\tau _{2}}(x+uh)) \\
&&-\mathbf{E}\Lambda _{A,p}(\sqrt{nh^{d}}\mathbf{z}_{N,\tau _{1}}(x;\eta
_{1}))\mathbf{E}\Lambda _{A^{\prime },p}(\sqrt{nh^{d}}\mathbf{z}_{N,\tau
_{2}}(x+uh;\eta _{2})),
\end{eqnarray*}%
and%
\begin{eqnarray*}
\Delta _{n,\tau _{1},\tau _{2},2}^{\eta }(x,u) &=&\mathbf{E}\Lambda _{A,p}(%
\sqrt{nh^{d}}\mathbf{z}_{N,\tau _{1}}(x))\Lambda _{A^{\prime },p}(\sqrt{%
nh^{d}}\mathbf{z}_{N,\tau _{2}}(x+uh)) \\
&&-\mathbf{E}\Lambda _{A,p}(\sqrt{nh^{d}}\mathbf{z}_{N,\tau _{1}}(x;\eta
_{1}))\Lambda _{A^{\prime },p}(\sqrt{nh^{d}}\mathbf{z}_{N,\tau
_{2}}(x+uh;\eta _{2})).
\end{eqnarray*}%
By H\"{o}lder inequality, for $C>0$ that depends only on $P$,%
\begin{equation*}
\left\vert \Delta _{n,\tau _{1},\tau _{2},2}^{\eta }(x,u)\right\vert \leq
CA_{1n}(x,u)+CA_{2n}(x,u)\text{,}
\end{equation*}%
where, if $p=1$ then we set $s=2$, and $q=1$, and if $p>1$, we set $%
s=(p+1)/(p-1)$ and $q=(1-1/s)^{-1},$%
\begin{eqnarray*}
A_{1n}(x,u) &=&(nh^{d})^{p}\left\{ \mathbf{E}\left\Vert \mathbf{z}_{N,\tau
_{1}}(x)-\mathbf{z}_{N,\tau _{1}}(x;\eta _{1})\right\Vert ^{2q}\right\} ^{%
\frac{1}{2q}} \\
&&\times \left( \left\{ \mathbf{E}\left\Vert \mathbf{z}_{N,\tau
_{1}}(x)\right\Vert ^{2s(p-1)}\right\} ^{\frac{1}{2s}}+\left\{ \mathbf{E}%
\left\Vert \mathbf{z}_{N,\tau _{1}}(x;\eta _{1})\right\Vert
^{2s(p-1)}\right\} ^{\frac{1}{2s}}\right) \\
&&\times \sqrt{\mathbf{E}\left( \left\Vert \mathbf{z}_{N,\tau
_{2}}(x+uh)\right\Vert ^{2p}\right) },
\end{eqnarray*}%
and%
\begin{eqnarray*}
A_{2n}(x,u) &=&(nh^{d})^{p}\left\{ \mathbf{E}\left\Vert \mathbf{z}_{N,\tau
_{2}}(x+uh)-\mathbf{z}_{N,\tau _{2}}(x+uh;\eta _{2})\right\Vert
^{2q}\right\} ^{\frac{1}{2q}} \\
&&\times \left( \left\{ \mathbf{E}\left\Vert \mathbf{z}_{N,\tau
_{2}}(x+uh)\right\Vert ^{2s(p-1)}\right\} ^{\frac{1}{2s}}+\left\{ \mathbf{E}%
\left\Vert \mathbf{z}_{N,\tau _{2}}(x+uh;\eta _{2})\right\Vert
^{2s(p-1)}\right\} ^{\frac{1}{2s}}\right) \\
&&\times \sqrt{\mathbf{E}\left( \left\Vert \mathbf{z}_{N,\tau _{1}}(x;\eta
_{1})\right\Vert ^{2p}\right) }.
\end{eqnarray*}%
Now,%
\begin{equation*}
\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\mathbf{E}\left\Vert 
\sqrt{nh^{d}}\{\mathbf{z}_{N,\tau }(x)-\mathbf{z}_{N,\tau }(x;\eta
_{1})\}\right\Vert ^{2q}=\mathbf{E}\left\Vert \sqrt{\bar{\varepsilon}}%
\mathbb{Z}\right\Vert ^{2q}=C\bar{\varepsilon}^{q}\text{,}
\end{equation*}%
where $\mathbb{Z}\in \mathbf{R}^{J}$ is a centered normal random vector with
identity covariance matrix $I_{J}$. Also, we deduce that for some $C>0$ that does not depend on $n$,
\begin{equation*}
\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\mathbf{E}\left\Vert 
\sqrt{nh^{d}}\mathbf{z}_{N,\tau }(x)\right\Vert ^{2s(p-1)}\leq C,
\end{equation*}%
by (\ref{ineqs}) of Lemma B5 and by the fact that $2s(p-1)=2(p+1)\leq M$.
Similarly, from some large $n$ on,%
\begin{eqnarray*}
&&\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\mathbf{E}\left(
\left\Vert \sqrt{nh^{d}}\mathbf{z}_{N,\tau }(x+uh;\eta _{2})\right\Vert
^{2p}\right) \\
&\leq &\sup_{\tau \in \mathcal{T},x\in \mathcal{S}_{\tau }(\varepsilon
)}\sup_{P\in \mathcal{P}}\mathbf{E}\left( \left\Vert \sqrt{nh^{d}}\mathbf{z}%
_{N,\tau }(x;\eta _{2})\right\Vert ^{2p}\right) <C,
\end{eqnarray*}%
for some $C>0$. Thus we conclude that for some $C>0$,%
\begin{equation*}
\sup_{(\tau _{1},\tau _{2})\in \mathcal{T}\times \mathcal{T}}\sup_{(x,u)\in (%
\mathcal{S}_{\tau _{1}}(\varepsilon )\cup \mathcal{S}_{\tau
_{2}}(\varepsilon ))\times \mathcal{U}}\sup_{P\in \mathcal{P}}\left(
A_{1n}(x,u)+A_{2n}(x,u)\right) \leq C\sqrt{\bar{\varepsilon}},
\end{equation*}%
and that for some $C>0,$%
\begin{equation*}
\sup_{(\tau _{1},\tau _{2})\in \mathcal{T}\times \mathcal{T}}\sup_{(x,u)\in (%
\mathcal{S}_{\tau _{1}}(\varepsilon )\cup \mathcal{S}_{\tau
_{2}}(\varepsilon ))\times \mathcal{U}}\sup_{P\in \mathcal{P}}\left\vert
\Delta _{n,\tau _{1},\tau _{2},2}^{\eta }(x,u)\right\vert \leq C\sqrt{\bar{%
\varepsilon}}.
\end{equation*}%
Using similar arguments, we also find that for some $C>0,$%
\begin{equation*}
\sup_{(\tau _{1},\tau _{2})\in \mathcal{T}\times \mathcal{T}}\sup_{(x,u)\in (%
\mathcal{S}_{\tau _{1}}(\varepsilon )\cup \mathcal{S}_{\tau
_{2}}(\varepsilon ))\times \mathcal{U}}\sup_{P\in \mathcal{P}}\left\vert
\Delta _{n,\tau _{1},\tau _{2},1}^{\eta }(x,u)\right\vert \leq C\sqrt{\bar{%
\varepsilon}}.
\end{equation*}%
Therefore, there exist $C_{1}>0$ and $C_{2}>0$ such that from some large $n$
on,%
\begin{eqnarray*}
&&\sup_{P\in \mathcal{P}}\left\vert \sigma _{n,A,A^{\prime },\bar{\varepsilon%
}}^{2}(B_{n},B_{n}^{\prime })-\sigma _{n,A,A^{\prime
}}^{2}(B_{n},B_{n}^{\prime })\right\vert \\
&\leq &C_{1}\sqrt{\bar{\varepsilon}}\int_{\mathcal{T}}\int_{\mathcal{T}%
}\int_{B_{n}}\int_{\mathcal{U}}w_{\tau _{1},B_{n}}(x)w_{\tau
_{2},B_{n}^{\prime }}(x+uh)dudxd\tau _{1}d\tau _{2}.
\end{eqnarray*}%
Since the last multiple integral is finite, we obtain the first statement of
Step 2.

We turn to the second statement of Step 2. Similarly as before, we write%
\begin{eqnarray*}
&&\left\vert \sigma _{n,A,A^{\prime },\bar{\varepsilon}}^{2}(B_{n},B_{n}^{%
\prime })-\sigma _{n,A,A^{\prime }}^{2}(B_{n},B_{n}^{\prime })\right\vert \\
&\leq &\int_{\mathcal{T}}\int_{\mathcal{T}}\int_{B_{n}}\int_{\mathcal{U}%
}\left\vert \Delta _{1,\tau _{1},\tau _{2}}^{\eta }(x,u)\right\vert w_{\tau
_{1},B_{n}}(x)w_{\tau _{2},B_{n}^{\prime }}(x+uh)dudxd\tau _{1}d\tau _{2} \\
&&+\int_{\mathcal{T}}\int_{\mathcal{T}}\int_{B_{n}}\int_{\mathcal{U}%
}\left\vert \Delta _{2,\tau _{1},\tau _{2}}^{\eta }(x,u)\right\vert w_{\tau
_{1},B_{n}}(x)w_{\tau _{2},B_{n}^{\prime }}(x+uh)dudxd\tau _{1}d\tau _{2},
\end{eqnarray*}%
where%
\begin{eqnarray*}
\Delta _{1,\tau _{1},\tau _{2}}^{\eta }(x,u) &=&\mathbf{E}\Lambda _{A,p}(%
\mathbb{W}_{n,\tau _{1},\tau _{2}}^{(1)}(x,u))\mathbf{E}\Lambda _{A^{\prime
},p}(\mathbb{W}_{n,\tau _{1},\tau _{2}}^{(2)}(x,u)) \\
&&-\mathbf{E}\Lambda _{A,p}(\mathbb{W}_{n,\tau _{1},\tau _{2},\bar{%
\varepsilon}}^{(1)}(x,u))\mathbf{E}\Lambda _{A^{\prime },p}(\mathbb{W}%
_{n,\tau _{1},\tau _{2},\bar{\varepsilon}}^{(2)}(x,u)),
\end{eqnarray*}%
and%
\begin{eqnarray*}
\Delta _{2,\tau _{1},\tau _{2}}^{\eta }(x,u) &=&\mathbf{E}\Lambda _{A,p}(%
\mathbb{W}_{n,\tau _{1},\tau _{2}}^{(1)}(x,u))\Lambda _{A^{\prime },p}(%
\mathbb{W}_{n,\tau _{1},\tau _{2}}^{(2)}(x,u)) \\
&&-\mathbf{E}\Lambda _{A,p}(\mathbb{W}_{n,\tau _{1},\tau _{2},\bar{%
\varepsilon}}^{(1)}(x,u))\Lambda _{A^{\prime },p}(\mathbb{W}_{n,\tau
_{1},\tau _{2},\bar{\varepsilon}}^{(2)}(x,u)).
\end{eqnarray*}%
Now, observe that for $C>0$ that does not depend on $\bar{\varepsilon},$ we
have by Lemma B1(i),%
\begin{equation*}
\sup_{(x,u)\in (\mathcal{S}_{\tau _{1}}(\varepsilon )\cup \mathcal{S}_{\tau
_{2}}(\varepsilon ))\times \mathcal{U}}\sup_{P\in \mathcal{P}}\left\Vert 
\tilde{\Sigma}_{n,\tau _{1},\tau _{2},\bar{\varepsilon}}^{1/2}(x,u)-\left[ 
\begin{array}{c}
\Sigma _{n,\tau _{1}}(x,0) \\ 
\Sigma _{n,\tau _{1},\tau _{2}}(x,u)%
\end{array}%
\begin{array}{c}
\Sigma _{n,\tau _{1},\tau _{2}}(x,u) \\ 
\Sigma _{n,\tau _{2}}(x+uh)%
\end{array}%
\right] ^{1/2}\right\Vert \leq C\sqrt{\bar{\varepsilon}}\text{.}
\end{equation*}%
Using this, recalling the definitions of $\mathbb{W}_{n,\tau _{1},\tau
_{2}}^{(1)}(x,u)$ and $\mathbb{W}_{n,\tau _{1},\tau _{2}}^{(2)}(x,u)$ in (%
\ref{Ws}), and following the previous arguments in the proof of Step 1, we obtain the second
statement of Step 2.
\end{proof}

\begin{LemmaC}
\label{lem-c7} \textit{Suppose that for some small }$\nu _{1}>0$, $%
n^{-1/2}h^{-d-\nu _{1}}\rightarrow 0$\textit{, as }$n\rightarrow \infty $%
\textit{\ and the conditions of Lemma B6 hold. Then there exists} $C>0$%
\textit{\ such that for any sequence of Borel sets }$B_{n}\subset \mathcal{S}%
,$ \textit{and }$A\subset \mathbb{N}_{J},$ \textit{from some large }$n$%
\textit{\ on,}%
\begin{eqnarray*}
&&\sup_{P\in \mathcal{P}}\mathbf{E}\left[ \left\vert
h^{-d/2}\int_{B_{n}}\left\{ \Lambda _{A,p}(\sqrt{nh^{d}}\mathbf{z}_{n,\tau
}(x))-\mathbf{E}\left[ \Lambda _{A,p}(\sqrt{nh^{d}}\mathbf{z}_{N,\tau }(x))%
\right] \right\} dQ(x,\tau )\right\vert \right] \\
&\leq &C\sqrt{Q(B_{n})}.
\end{eqnarray*}
\end{LemmaC}

\begin{rem}
The result is in the same spirit as Lemma 6.2 of Gin\'{e}, Mason, and
Zaitsev (2003). (Also see Lemma A8 of Lee, Song and Whang (2013).) However,
unlike these results, the location normalization here involves\ $\mathbf{E}%
[\Lambda _{A,p}(\sqrt{nh^{d}}\mathbf{z}_{N,\tau }(x))]$\ instead of $\mathbf{%
E}[\Lambda _{A,p}(\sqrt{nh^{d}}\mathbf{z}_{n,\tau }(x))]$. We can obtain the
same result with $\mathbf{E}[\Lambda _{A,p}(\sqrt{nh^{d}}\mathbf{z}_{N,\tau
}(x))]$ replaced by $\mathbf{E}[\Lambda _{A,p}(\sqrt{nh^{d}}\mathbf{z}%
_{n,\tau }(x))]$, but with a stronger bandwidth condition.
\end{rem}

Like Lemma B6, the result of Lemma B7 does not require that the quantities $%
\sqrt{nh^{d}}\mathbf{z}_{n,\tau }(x)$ and $\sqrt{nh^{d}}\mathbf{z}_{N,\tau
}(x)$ have a (pointwise in $x$) nondegenerate limit distribution.

\begin{proof}[Proof of Lemma B\protect\ref{lem-c7}]
As in the proof of Lemma A8 of \citeasnoun{LSW}, it suffices to show that
there exists $C>0$ such that $C$ does not depend on $n$ and for any Borel
set $B_n\subset \mathbf{R}$,\newline
\textbf{Step 1:}%
\begin{equation*}
\sup_{P\in \mathcal{P}}\mathbf{E}\left[ \left\vert
h^{-d/2}\int_{B_{n}}\left\{ \Lambda _{A,p}(\sqrt{nh^{d}}\mathbf{z}_{n,\tau
}(x))-\Lambda _{A,p}(\sqrt{nh^{d}}\mathbf{z}_{N,\tau }(x))\right\} dQ(x,\tau
)\right\vert \right] \leq CQ(B_{n}),\text{ and}
\end{equation*}%
\textbf{Step 2:}%
\begin{equation*}
\sup_{P\in \mathcal{P}}\mathbf{E}\left[ \left\vert
h^{-d/2}\int_{B_{n}}\left\{ \Lambda _{A,p}(\sqrt{nh^{d}}\mathbf{z}_{N,\tau
}(x))-\mathbf{E}\left[ \Lambda _{A,p}(\sqrt{nh^{d}}\mathbf{z}_{N,\tau }(x))%
\right] \right\} dQ(x,\tau )\right\vert \right] \leq C\sqrt{Q(B_{n})}.
\end{equation*}

By chaining Steps 1 and 2, we obtain the desired result.\newline
\textbf{Proof of Step 1: }Similarly as in (2.13) of \citeasnoun{Horvath:91},
we first write%
\begin{equation}
\mathbf{z}_{n,\tau }(x)=\mathbf{z}_{N,\tau }(x)+\mathbf{v}_{n,\tau }(x)+%
\mathbf{s}_{n,\tau }(x),  \label{dec55}
\end{equation}%
where, for $\beta _{n,x,\tau }(Y_{i},(X_{i}-x)/h)$ defined prior to Lemma B5,%
\begin{eqnarray*}
\mathbf{v}_{n,\tau }(x) &\equiv &\left( \frac{n-N}{n}\right) \cdot \frac{1}{%
h^{d}}\mathbf{E}\left[ \beta _{n,x,\tau }\left( Y_{i},\frac{X_{i}-x}{h}%
\right) \right] \text{ and} \\
\mathbf{s}_{n,\tau }(x) &\equiv &\frac{1}{nh^{d}}\sum_{i=N+1}^{n}\left\{
\beta _{n,x,\tau }\left( Y_{i},\frac{X_{i}-x}{h}\right) -\mathbf{E}\left[
\beta _{n,x,\tau }\left( Y_{i},\frac{X_{i}-x}{h}\right) \right] \right\} ,
\end{eqnarray*}%
and we write $\sum_{i=N+1}^{n}=0$ if $N=n$, and $\sum_{i=N+1}^{n}=-
\sum_{i=n+1}^{N}$ if $N>n$.

Using (\ref{dec55}), we deduce that for some $C_{1},C_{2}>0$ that depend
only on $p$,%
\begin{eqnarray}
&&\int_{B_{n}}\left\vert \Lambda _{A,p}\left( \mathbf{z}_{n,\tau }(x)\right)
-\Lambda _{A,p}\left( \mathbf{z}_{N,\tau }(x)\right) \right\vert dQ(x,\tau )
\label{ineq3} \\
&\leq &C_{1}\int_{B_{n}}\left\Vert \mathbf{v}_{n,\tau }(x)\right\Vert \left(
\left\Vert \mathbf{z}_{n,\tau }(x)\right\Vert ^{p-1}+\left\Vert \mathbf{z}%
_{N,\tau }(x)\right\Vert ^{p-1}\right) dQ(x,\tau )  \notag \\
&&+C_{2}\int_{B_{n}}\left\Vert \mathbf{s}_{n,\tau }(x)\right\Vert \left(
\left\Vert \mathbf{z}_{n,\tau }(x)\right\Vert ^{p-1}+\left\Vert \mathbf{z}%
_{N,\tau }(x)\right\Vert ^{p-1}\right) dQ(x,\tau )  \notag \\
&\equiv &D_{1n}+D_{2n}\text{, say.}  \notag
\end{eqnarray}%
To deal with $D_{1n}$ and $D_{2n}$, we first show the following:\medskip

\noindent \textsc{Claim 1:} $\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in 
\mathcal{P}}\mathbf{E}[||\mathbf{v}_{n,\tau }(x)||^{2}]=O(n^{-1})$,
and\medskip

\noindent \textsc{Claim 2:} $\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in 
\mathcal{P}}\mathbf{E}[||\mathbf{s}_{n,\tau }(x)||^{2}]=O(n^{-3/2}h^{-d})$%
.\medskip

\noindent \textsc{Proof of Claim 1:} First, note that%
\begin{equation*}
\sup_{(x,\tau )\in \mathcal{S}}\mathbf{E}\left[ ||\mathbf{v}_{n,\tau
}(x)||^{2}\right] \leq \mathbf{E}\left\vert \frac{n-N}{n}\right\vert
^{2}\cdot \sup_{(x,\tau )\in \mathcal{S}}\left\Vert \frac{1}{h^{d}}\mathbf{E}%
\left[ \beta _{n,x,\tau }\left( Y_{i},\frac{X_{i}-x}{h}\right) \right]
\right\Vert ^{2}.
\end{equation*}

Since $\mathbf{E}|n^{-1/2}(n-N)|^{2}$ does not depend on the joint
distribution of $(Y_{i},X_{i})$, $\mathbf{E}|n^{-1/2}(n-N)|^{2}\leq O(1)$
uniformly over $P\in \mathcal{P}$. Combining this with the second statement
of (\ref{ineqs}), the product on the right hand side becomes $O(n^{-1})$
uniformly over $P\in \mathcal{P}$.

\noindent \textsc{Proof of Claim 2:} Let $\eta _{1}\in \mathbf{R}^{J}$ be
the random vector defined prior to Lemma B6, and define%
\begin{equation*}
\mathbf{s}_{n,\tau }(x;\eta _{1})\equiv \mathbf{s}_{n,\tau }(x)+\frac{%
(N-n)\eta _{1}}{n^{3/2}h^{d/2}}.
\end{equation*}%
Note that%
\begin{equation}
\mathbf{E}\left\Vert \mathbf{s}_{n,\tau }(x)\right\Vert ^{2}\leq 2\mathbf{E}%
\left\Vert \mathbf{s}_{n,\tau }(x;\eta _{1})\right\Vert ^{2}+\frac{2}{%
n^{2}h^{d}}\mathbf{E}\left\Vert \frac{(N-n)\eta _{1}}{\sqrt{n}}\right\Vert
^{2}.  \label{dec22}
\end{equation}%
As for the last term, since $N$ and $\eta _{1}$ are independent, it is
bounded by%
\begin{equation*}
\frac{1}{n^{2}h^{d}}\left( \mathbf{E}\left\vert \frac{N-n}{\sqrt{n}}%
\right\vert ^{2}\right) \cdot \mathbf{E}\left\Vert \eta _{1}\right\Vert
^{2}\leq \frac{C\bar{\varepsilon}}{n^{2}h^{d}}=O(n^{-2}h^{-d-\nu _{1}}),
\end{equation*}%
from some large $n$ on.

As for the leading expectation on the right hand side of (\ref{dec22}), we
write%
\begin{eqnarray*}
\mathbf{E}\left\Vert \sqrt{nh^{d}}\mathbf{s}_{n,\tau }(x;\eta
_{1})\right\Vert ^{2} &=&\mathbf{E}\left\Vert \frac{1}{\sqrt{n}}%
\sum_{i=N+1}^{n}q_{n,\tau ,1}^{(i)}(x)\right\Vert ^{2} \\
&=&\frac{1}{n}\sum_{j=1}^{J}\bar{\sigma}_{n,\tau ,j}^{2}(x)\mathbf{E}\left(
\sum_{i=N+1}^{n}\frac{q_{n,\tau ,1,j}^{(i)}(x)}{\bar{\sigma}_{n,\tau ,j}(x)}%
\right) ^{2},
\end{eqnarray*}%
where $q_{n,\tau ,1}^{(i)}(x)$'s ($i=1,2,...$) are i.i.d.
copies of $q_{n,\tau }(x)+\eta _{1}$ and $q_{n,\tau ,1,j}^{(i)}(x)$ is the $%
j $-th entry of $q_{n,\tau ,1}^{(i)}(x),$ and $\bar{\sigma}_{n,\tau
,j}^{2}(x)\equiv Var(q_{n,\tau ,1,j}^{(i)}(x)).$ Recall that $q_{n,\tau }(x)$
was defined prior to Lemma B5. Now we apply Lemma 1(i) of %
\citeasnoun{Horvath:91} to deduce that%
\begin{eqnarray*}
&&\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\mathbf{E}\left(
\sum_{i=N+1}^{n}\frac{q_{n,\tau ,1,j}^{(i)}(x)}{\bar{\sigma}_{n,\tau ,j}(x)}%
\right) ^{2} \\
&\leq &\mathbf{E}|N-n|\cdot \mathbf{E}|\mathbb{Z}_{1}|^{2}+C\mathbf{E}%
\left\vert N-n\right\vert ^{1/2}\cdot \sup_{(x,\tau )\in \mathcal{S}%
}\sup_{P\in \mathcal{P}}\mathbf{E}\left\vert \frac{q_{n,\tau ,1,j}^{(i)}(x)}{%
\bar{\sigma}_{n,\tau ,j}(x)}\right\vert ^{3} \\
&&+C\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\mathbf{E}%
\left\vert \frac{q_{n,\tau ,1,j}^{(i)}(x)}{\bar{\sigma}_{n,\tau ,j}(x)}%
\right\vert ^{4},
\end{eqnarray*}%
for some $C>0$, where $\mathbb{Z}_{1}\sim N(0,1)$.

First, observe that $\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}%
\bar{\sigma}_{n,\tau ,j}(x)<\infty $ by Lemma B5, and 
\begin{equation}
\inf_{(x,\tau )\in \mathcal{S}}\inf_{P\in \mathcal{P}}\bar{\sigma}_{n,\tau
,j}(x)>\bar{\varepsilon}>0,  \label{lb}
\end{equation}%
due to the additive term $\eta _{1}$ in $q_{n,\tau }(x)+\eta _{1}$. Let $%
\eta _{1j}$ be the $j$-th entry of $\eta _{1}$. We apply Lemma B5 to deduce
that for some $C>0,$ from some large $n$ on,%
\begin{eqnarray}
\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\mathbf{E}|(q_{n,\tau
,j}(x)+\eta _{1j})/\bar{\sigma}_{n,\tau ,j}(x)|^{3} &\leq &Ch^{-(d/2)-(\nu
_{1}/2)}\text{ and}  \label{arg1} \\
\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\mathbf{E}|(q_{n,\tau
,j}(x)+\eta _{1j})/\bar{\sigma}_{n,\tau ,j}(x)|^{4} &\leq &Ch^{-d-\nu _{1}}.
\notag
\end{eqnarray}%
Since $\mathbf{E}|N-n|=O(n^{1/2})$ and $\mathbf{E}\left\vert N-n\right\vert
^{1/2}=O(n^{1/4})$ (e.g. (2.21) and (2.22) of \citeasnoun{Horvath:91}),
there exists $C>0$ such that%
\begin{equation}
\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\mathbf{E}\left(
\sum_{i=N+1}^{n}\frac{q_{n,\tau ,1,j}^{(i)}(x)}{\bar{\sigma}_{n,\tau ,j}(x)}%
\right) ^{2}\leq \frac{C}{\bar{\varepsilon}^{4}}\left\{
n^{1/2}+n^{1/4}h^{-(d/2)-(\nu _{1}/2)}+h^{-d-\nu _{1}}\right\} .
\label{arg2}
\end{equation}%
This implies that for some $C>0,$ (with $\bar{\varepsilon}>0$ fixed while $%
n\rightarrow \infty $)%
\begin{eqnarray}
&&\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\mathbf{E}\left\Vert 
\sqrt{nh^{d}}\mathbf{s}_{n,\tau }(x)\right\Vert ^{2}  \label{arg3} \\
&\leq &O\left( n^{-1}h^{-\nu _{1}}\right) +O\left(
n^{-1/2}+n^{-3/4}h^{-(d/2)-(\nu _{1}/2)}+n^{-1}h^{-d-\nu _{1}}\right)  \notag
\\
&=&O\left( n^{-1}h^{-\nu _{1}}\right) +O(n^{-1/2})=O(n^{-1/2}),  \notag
\end{eqnarray}%
since $n^{-1/2}h^{-d-\nu _{1}}\rightarrow 0$. Hence, we obtain Claim 2.

Using Claim 1 and the second statement of Lemma B5, we deduce that%
\begin{eqnarray*}
\sup_{P\in \mathcal{P}}\mathbf{E}\left[ n^{p/2}h^{d(p-1)/2}D_{1n}\right]
&\leq &C_{1}Q(B_{n})\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}%
\sqrt{\mathbf{E}\left\Vert \sqrt{n}\mathbf{v}_{n,\tau }(x)\right\Vert ^{2}}
\\
&&\times \sqrt{\mathbf{E}\left\Vert \sqrt{nh^{d}}\mathbf{z}_{n,\tau
}(x)\right\Vert ^{2p-2}+\mathbf{E}\left\Vert \sqrt{nh^{d}}\mathbf{z}_{N,\tau
}(x)\right\Vert ^{2p-2}} \\
&\leq &C_{2}Q(B_{n}),
\end{eqnarray*}%
for $C_{1},C_{2}>0$. Similarly, we can see that 
\begin{equation*}
\sup_{P\in \mathcal{P}}\mathbf{E}\left[ n^{p/2}h^{d(p-1)/2}D_{2n}\right]
=O(n^{-1/2}h^{-d})=o(1),
\end{equation*}%
using Claim 2 and the second statement of Lemma B5. Thus, we obtain Step 1.%
\newline
\textbf{Proof of Step 2:} We can follow the proof of Lemma B6 to show that%
\begin{eqnarray*}
&&\mathbf{E}\left[ h^{-d/2}\int_{B_{n}}\left( \Lambda _{A,p}(\sqrt{nh^{d}}%
\mathbf{z}_{N,\tau }(x))-\mathbf{E}\left[ \Lambda _{A,p}(\sqrt{nh^{d}}%
\mathbf{z}_{N,\tau }(x))\right] \right) dQ(x,\tau )\right] ^{2} \\
&=&\int_{\mathcal{T}}\int_{\mathcal{T}}\int_{B_{n,\tau _{1}}\cap B_{n,\tau
_{2}}}\int_{\mathcal{U}}C_{n,\tau _{1},\tau _{2},A,A^{\prime
}}(x,u)dudxd\tau _{1}d\tau _{2}+o(1),
\end{eqnarray*}%
where $B_{n,\tau}$ is the $\tau$-section of $B_n$ defined at the beginning of the proof of Lemma B6, $C_{n,\tau _{1},\tau _{2},A,A^{\prime }}(x,u)$ is defined in (\ref{C}),
and the last $o(1)$ term is $o(1)$ uniform over $P\in \mathcal{P}$. Now, observe that%
\begin{eqnarray*}
&&\sup_{(\tau _{1},\tau _{2})\in \mathcal{T}\times \mathcal{T}}\sup_{u\in 
\mathcal{U}}\sup_{x\in \mathcal{X}}\sup_{P\in \mathcal{P}}\left\vert
C_{n,\tau _{1},\tau _{2},A,A^{\prime }}(x,u)\right\vert \\
&\leq &\sup_{(\tau _{1},\tau _{2})\in \mathcal{T}\times \mathcal{T}%
}\sup_{u\in \mathcal{U}}\sup_{x\in \mathcal{X}}\sup_{P\in \mathcal{P}}\sqrt{%
\mathbf{E||}\mathbb{W}_{n,\tau _{1},\tau _{2}}^{(1)}(x,u)||^{2p}\mathbf{E||}%
\mathbb{W}_{n,\tau _{1},\tau _{2}}^{(2)}(x,u)||^{2p}}<\infty .
\end{eqnarray*}%
Therefore,%
\begin{eqnarray*}
&&\mathbf{E}\left[ \left\vert h^{-d/2}\int_{B_{n}}\left( \Lambda _{A,p}(%
\sqrt{nh^{d}}\mathbf{z}_{N,\tau }(x))-\mathbf{E}\left[ \Lambda _{A,p}(\sqrt{%
nh^{d}}\mathbf{z}_{N,\tau }(x))\right] \right) dQ(x,\tau )\right\vert \right]
\\
&\leq &\sqrt{\int_{\mathcal{T}}\int_{\mathcal{T}}\int_{\mathcal{U}%
}\int_{B_{n,\tau _{1}}\cap B_{n,\tau _{2}}}C_{n,\tau _{1},\tau
_{2},A,A^{\prime }}(x,u)dxdud\tau _{1}d\tau _{2}}+o(1) \\
&\leq &C\sqrt{\int_{\mathcal{T}}\int_{\mathcal{T}}\int_{\mathcal{U}%
}\int_{B_{n,\tau _{1}}\cap B_{n,\tau _{2}}}dxdud\tau _{1}d\tau _{2}}+o(1),
\end{eqnarray*}%
for some $C>0.$ Now, observe that%
\begin{equation*}
\int_{\mathcal{T}}\int_{\mathcal{T}}\int_{B_{n,\tau _{1}}\cap B_{n,\tau
_{2}}}dxd\tau _{1}d\tau _{2}\leq \int_{\mathcal{T}}d\tau _{2}\cdot \left(
\int_{\mathcal{T}}\int_{B_{n,\tau _{1}}}dxd\tau _{1}\right) \leq CQ(B_{n}),
\end{equation*}%
because $\mathcal{T}$ is a bounded set. Thus the proof of Step 2 is
completed.
\end{proof}

The next lemma shows the joint asymptotic normality of a Poissonized version
of a normalized test statistic and a Poisson random variable. Using this
result, we can apply the de-Poissonization lemma in Lemma B3. To define a
Poissonized version of a normalized test statistic, we introduce some
notation.

Let $\mathcal{C}\subset \mathbf{R}^{d}$ be a compact set such that $\mathcal{%
C}$ does not depend on $P\in \mathcal{P}$ and $\alpha _{P}\equiv P\{X\in 
\mathbf{R}^{d}\backslash \mathcal{C}\}$ satisfies that $0<\inf_{P\in 
\mathcal{P}}\alpha _{P}\leq \sup_{P\in \mathcal{P}}\alpha _{P}<1$. Existence
of such $\mathcal{C}$ is assumed in Assumption A6(ii). For $c_{n}\rightarrow
\infty $, we let $B_{n,A}(c_{n};\mathcal{C})\equiv B_{n,A}(c_{n})\cap (%
\mathcal{C}\times \mathcal{T})$, where we recall the definition of $%
B_{n,A}(c_{n})=B_{n,A}(c_{n},c_{n})$. (Recall the definition of $%
B_{n,A}(c_{n,1},c_{n,2})$ before Lemma 1.) Define%
\begin{eqnarray*}
\zeta _{n,A} &\equiv &\int_{B_{n,A}(c_{n};\mathcal{C})}\Lambda _{A,p}(\sqrt{%
nh^{d}}\mathbf{z}_{n,\tau }(x))dQ(x,\tau ),\text{ and} \\
\zeta _{N,A} &\equiv &\int_{B_{n,A}(c_{n};\mathcal{C})}\Lambda _{A,p}(\sqrt{%
nh^{d}}\mathbf{z}_{N,\tau }(x))dQ(x,\tau ).
\end{eqnarray*}%
Let $\mu _{A}$'s be real numbers indexed by $A\in \mathcal{N}_{J}$, and
define%
\begin{equation*}
\sigma _{n}^{2}(\mathcal{C})\equiv \sum_{A\in \mathcal{N}_{J}}\sum_{A^{%
\prime }\in \mathcal{N}_{J}}\mu _{A}\mu _{A^{\prime }}\sigma _{n,A,A^{\prime
}}(B_{n,A}(c_{n};\mathcal{C}),B_{n,A^{\prime }}(c_{n};\mathcal{C}))\text{%
\textit{,}}
\end{equation*}%
\textit{\ }where we recall the definition of $\sigma _{n,A,A^{\prime
}}(\cdot ,\cdot )$ prior to Lemma B6. Define%
\begin{equation*}
S_{n}\equiv h^{-d/2}\sum_{A\in \mathcal{N}_{J}}\mu _{A}\left\{ \zeta _{N,A}-%
\mathbf{E}\zeta _{N,A}\right\} .
\end{equation*}%
Also define%
\begin{eqnarray*}
U_{n} &\equiv &\frac{1}{\sqrt{n}}\left\{ \sum_{i=1}^{N}1\{X_{i}\in \mathcal{C%
}\}-nP\left\{ X_{i}\in \mathcal{C}\right\} \right\} ,\text{ and} \\
V_{n} &\equiv &\frac{1}{\sqrt{n}}\left\{ \sum_{i=1}^{N}1\{X_{i}\in \mathbf{R}%
^{d}\backslash \mathcal{C}\}-nP\left\{ X_{i}\in \mathbf{R}^{d}\backslash 
\mathcal{C}\right\} \right\} .
\end{eqnarray*}%
Let 
\begin{equation*}
H_{n}\equiv \left[ \frac{S_{n}}{\sigma _{n}(\mathcal{C})},\frac{U_{n}}{\sqrt{%
1-\alpha _{P}}}\right] ^{\top }.
\end{equation*}%
The following lemma establishes the joint convergence of $H_{n}$. In doing
so, we need to be careful in dealing with uniformity in $P\in \mathcal{P}$,
and potential degeneracy of the normalized test statistic $S_{n}$.

\begin{LemmaC}
\label{lem-c8} \textit{Suppose that the conditions of Lemma B7 hold and that 
}$c_{n}\rightarrow \infty $ \textit{as }$n\rightarrow \infty .$

\noindent (i) If $\liminf_{n\rightarrow \infty }\inf_{P\in \mathcal{P}%
}\sigma _{n}^{2}(\mathcal{C})>0$, \textit{then}%
\begin{equation*}
\sup_{P\in \mathcal{P}}\sup_{t\in \mathbf{R}^{2}}\left\vert P\left\{
H_{n}\leq t\right\} -P\left\{ \mathbb{Z}\leq t\right\} \right\vert
\rightarrow 0,
\end{equation*}%
\textit{where }$\mathbb{Z}\sim N(0,I_{2})$.

\noindent (ii) If $\limsup_{n\rightarrow \infty }\sigma _{n}^{2}(\mathcal{C}%
)=0$,\textit{\ then for each }$(t_{1},t_{2})\in \mathbf{R}^{2},$%
\begin{equation*}
\left\vert P\left\{ S_{n}\leq t_{1}\text{ and }\frac{U_{n}}{\sqrt{1-\alpha
_{P}}}\leq t_{2}\right\} -1\{0\leq t_{1}\}P\left\{ \mathbb{Z}_{1}\leq
t_{2}\right\} \right\vert \rightarrow 0,
\end{equation*}%
\textit{where }$\mathbb{Z}_{1}\sim N(0,1)$.
\end{LemmaC}

\begin{rem}
The joint convergence result in Lemma B8 is divided into two separate results. The first
case is a situation where $S_{n}$ is asymptotically nondegenerate uniformly
in $P\in \mathcal{P}$. The second case deals with a situation where $S_{n}$
is asymptotically degenerate for some $P\in \mathcal{P}$.
\end{rem}

\begin{proof}[Proof of Lemma B\protect\ref{lem-c8}]
(i) Choose any small $\bar{\varepsilon}>0$ and let%
\begin{equation*}
H_{n,\bar{\varepsilon}}\equiv \left[ \frac{S_{n,\bar{\varepsilon}}}{\sigma
_{n,\bar{\varepsilon}}(\mathcal{C})},\frac{U_{n}}{\sqrt{1-\alpha _{P}}}%
\right] ^{\top },
\end{equation*}%
where $S_{n,\bar{\varepsilon}}$ is equal to $S_{n}$, except that $\zeta
_{N,A}$ is replaced by%
\begin{equation*}
\zeta _{N,A,\bar{\varepsilon}}\equiv \int_{B_{n,A}(c_{n};\mathcal{C}%
)}\Lambda _{A,p}(\sqrt{nh^{d}}\mathbf{z}_{N,\tau }(x;\eta _{1}))dQ(x,\tau ),
\end{equation*}%
and $\mathbf{z}_{N,\tau }(x;\eta _{1})$ is as defined prior to Lemma B6, and 
$\sigma _{n,\bar{\varepsilon}}(\mathcal{C})$ is $\sigma _{n}(\mathcal{C})$
except that $\tilde{\Sigma}_{n,\tau _{1},\tau _{2}}(x,u)$ is replaced by $%
\tilde{\Sigma}_{n,\tau _{1},\tau _{2},\bar{\varepsilon}}(x,u)$. Also let 
\begin{equation*}
C_{n}\equiv \mathbf{E}H_{n}H_{n}^{\top }\text{ and }C_{n,\bar{\varepsilon}%
}\equiv \mathbf{E}H_{n,\bar{\varepsilon}}H_{n,\bar{\varepsilon}}^{\top }%
\text{.}
\end{equation*}%
First, we show the following statements.\medskip \newline
\textbf{Step 1:} For some $C>0$, $\sup_{P\in \mathcal{P}}|Cov(S_{n,\bar{%
\varepsilon}}-S_{n},U_{n})|\leq C\sqrt{\bar{\varepsilon}}$, for each fixed $%
\bar{\varepsilon}>0$.\medskip \newline
\textbf{Step 2: }$\sup_{P\in \mathcal{P}}\left\vert Cov(S_{n,\bar{\varepsilon%
}},U_{n})\right\vert =o(h^{d/2}),$ as $n\rightarrow \infty .$\medskip 
\newline
\textbf{Step 3: }There exists $c>0$ such that from some large $n$ on,%
\begin{equation*}
\inf_{P\in \mathcal{P}}\lambda _{\min }(C_{n})>c.
\end{equation*}%
\textbf{Step 4: }As $n\rightarrow \infty ,$%
\begin{equation*}
\sup_{P\in \mathcal{P}}\sup_{t\in \mathbf{R}^{2}}\left\vert P\left\{
C_{n}^{-1/2}H_{n}\leq t\right\} \rightarrow P\left\{ \mathbb{Z}\leq
t\right\} \right\vert \rightarrow 0\text{.}
\end{equation*}%
\newline
From Steps 1-3, we find that $\sup_{P\in \mathcal{P}}\left\Vert
C_{n}-I_{2}\right\Vert \rightarrow 0$, as $n\rightarrow \infty $ and as $%
\bar{\varepsilon}\rightarrow 0$. By Step 4, we obtain (i) of Lemma
B8.\medskip \newline
\textbf{Proof of Step 1: }Observe that from an inequality similar to (\ref%
{ineq3}) in the proof of Lemma B7, 
\begin{equation*}
\left\vert \zeta _{N,A,\bar{\varepsilon}}-\zeta _{N,A}\right\vert \leq
C||\eta _{1}||\int_{B_{n,A}(c_{n};\mathcal{C})}\left\Vert \sqrt{nh^{d}}%
\mathbf{z}_{N,\tau }(x)\right\Vert ^{p-1}dQ(x,\tau ).
\end{equation*}%
Using the fact that $\mathcal{S}$ is compact and does not depend on $P\in 
\mathcal{P}$, for some constants $C_{1},C_{2},C_{3}>0$ that do not depend on 
$P\in \mathcal{P}$ or $n$, 
\begin{eqnarray*}
\mathbf{E}\left\vert \zeta _{N,A,\bar{\varepsilon}}-\zeta _{N,A}\right\vert
^{2} &\leq &C_{1}\mathbf{E}\left[ ||\eta _{1}||^{2}\right] \cdot
\int_{B_{n,A}(c_{n};\mathcal{C})}\mathbf{E}\left\Vert \sqrt{nh^{d}}\mathbf{z}%
_{N,\tau }(x)\right\Vert ^{2p-2}dQ(x,\tau ) \\
&\leq &C_{2}\bar{\varepsilon}\cdot \int_{B_{n,A}(c_{n};\mathcal{C})}\mathbf{E%
}\left\Vert \sqrt{nh^{d}}\mathbf{z}_{N,\tau }(x)\right\Vert ^{2p-2}dQ(x,\tau
)\leq C_{3}\bar{\varepsilon},
\end{eqnarray*}%
by the independence between $\eta _{1}$ and $\{\mathbf{z}_{N,\tau
}(x):(x,\tau )\in \mathcal{S}\}$, and by the second statement of Lemma B5.
From the fact that 
\begin{equation*}
\sup_{P\in \mathcal{P}}\mathbf{E}U_{n}^{2}\leq \sup_{P\in \mathcal{P}%
}(1-\alpha _{P})\leq 1,
\end{equation*}%
we obtain the desired result.\medskip \newline
\textbf{Proof of Step 2: }Let $\Sigma _{2n,\tau ,\bar{\varepsilon}}$ be the
covariance matrix of $[(q_{n,\tau }(x)+\eta _{1})^{\top },\tilde{U}%
_{n}]^{\top }$, where $\tilde{U}_{n}=U_{n}/\sqrt{P\{X\in \mathcal{C}\}}$ and $q_{n,\tau}(x)$ was defined prior to Lemma B5. We
can write $\Sigma _{2n,\tau ,\bar{\varepsilon}}$ as%
\begin{eqnarray*}
&&\left[ 
\begin{array}{c}
\Sigma _{n,\tau ,\tau }(x,0)+\bar{\varepsilon}I_{J} \\ 
\mathbf{E}[(q_{n,\tau }(x)+\eta _{1})^{\top }\tilde{U}_{n}]%
\end{array}%
\begin{array}{c}
\mathbf{E}[(q_{n,\tau }(x)+\eta _{1})\tilde{U}_{n}] \\ 
1%
\end{array}%
\right] \\
&=&\left[ 
\begin{array}{c}
\Sigma _{n,\tau ,\tau }(x,0) \\ 
\sqrt{1-\bar{\varepsilon}}\mathbf{E}[q_{n,\tau }^{\top }(x)\tilde{U}_{n}]%
\end{array}%
\begin{array}{c}
\sqrt{1-\bar{\varepsilon}}\mathbf{E}[q_{n,\tau }(x)\tilde{U}_{n}] \\ 
1-\bar{\varepsilon}%
\end{array}%
\right] +\left[ 
\begin{array}{c}
\bar{\varepsilon}I_{J} \\ 
\mathbf{0}^{\top }%
\end{array}%
\begin{array}{c}
\mathbf{0} \\ 
\bar{\varepsilon}%
\end{array}%
\right] +A_{n,\tau }(x),
\end{eqnarray*}%
where%
\begin{equation*}
A_{n,\tau }(x)\equiv \left[ 
\begin{array}{c}
\mathbf{0} \\ 
\left( 1-\sqrt{1-\bar{\varepsilon}}\right) \mathbf{E}[q_{n,\tau }^{\top }(x)%
\tilde{U}_{n}]%
\end{array}%
\begin{array}{c}
\left( 1-\sqrt{1-\bar{\varepsilon}}\right) \mathbf{E}[q_{n,\tau }(x)\tilde{U}%
_{n}] \\ 
\mathbf{0}%
\end{array}%
\right] .
\end{equation*}%
The first matrix on the right hand side is certainly positive semidefinite.
Note that 
\begin{equation*}
\left( q_{n,\tau ,j}(x),\tilde{U}_{n}\right) \overset{d}{=}\left( \frac{1}{%
\sqrt{n}}\sum_{k=1}^{n}q_{n,\tau ,j}^{(k)}(x),\frac{1}{\sqrt{n}}%
\sum_{k=1}^{n}\tilde{U}_{n}^{(k)}\right) ,
\end{equation*}%
where $(q_{n,\tau ,j}^{(k)}(x),\tilde{U}_{n}^{(k)})$'s with $k=1,...,n$ are i.i.d. copies of $(q_{n,\tau ,j}(x),\bar{U}_{n})$, where 
\begin{equation*}
\bar{U}_{n}\equiv \frac{1}{\sqrt{P\{X\in \mathcal{C}\}}}\left\{ \sum_{1\leq
i\leq N_{1}}1\{X_{i}\in \mathcal{C}\}-P\left\{ X_{i}\in \mathcal{C}\right\}
\right\} ,
\end{equation*}%
where $N_{1}$ is the Poisson random variable with mean $1$ that is involved
in the definition of $q_{n,\tau ,j}(x)$. Hence as for $A_{n,\tau }(x)$, note
that for $C_{1},C_{2}>0,$%
\begin{eqnarray}
\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\left\vert \mathbf{E}%
\left[ q_{n,\tau ,j}(x)\tilde{U}_{n}\right] \right\vert &\leq &\sup_{(x,\tau
)\in \mathcal{S}}\sup_{P\in \mathcal{P}}\left\vert \mathbf{E}\left[
q_{n,\tau ,j}^{(k)}(x)\tilde{U}_{n}^{(k)}\right] \right\vert  \label{dev7} \\
&\leq &\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\frac{\mathbf{E}%
\left[ |q_{n,\tau ,j}(x)|\right] }{\sqrt{P\{X_{i}\in \mathcal{C}\}}}  \notag
\\
&\leq &\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}} \frac{C_{1}h^{d}%
k_{n,\tau ,j,1}(x)}{h^{d/2}\left( 1-\alpha _{P}\right) }\leq C_{2}h^{d/2},
\notag
\end{eqnarray}
where $k_{n,\tau,j,1}(x)$ was defined prior to Lemma B4. We conclude that%
\begin{equation*}
\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}||A_{n,\tau
}(x)||=O(h^{d/2}).
\end{equation*}%
Therefore, from some large $n$ on,%
\begin{equation}
\inf_{(x,\tau )\in \mathcal{S}}\inf_{P\in \mathcal{P}}\lambda _{\min }\left(
\Sigma _{2n,\tau ,\bar{\varepsilon}}\right) \geq \bar{\varepsilon}/2.
\label{LB2}
\end{equation}

Let 
\begin{equation*}
W_{n,\tau }(x;\eta _{1})\equiv \Sigma _{2n,\tau ,\bar{\varepsilon}}^{-1/2}%
\left[ 
\begin{array}{c}
q_{n,\tau }(x)+\eta _{1} \\ 
\tilde{U}_{n}%
\end{array}%
\right] .
\end{equation*}%
Similarly as in (\ref{der5}), we find that for some $C>0$, from some large $%
n $ on,%
\begin{eqnarray*}
&&\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\mathbf{E}\left\Vert
W_{n,\tau }(x;\eta _{1})\right\Vert ^{3} \\
&\leq &C\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\lambda _{\max
}^{3/2}\left( \Sigma _{2n,\tau ,\bar{\varepsilon}}^{-1}\right) \sup_{(x,\tau
)\in \mathcal{S}}\sup_{P\in \mathcal{P}}\left\{ \mathbf{E}\left[ ||q_{n,\tau
}(x)+\eta _{1}||^{3}\right] +\mathbf{E}\left[ |\tilde{U}_{n}|^{3}\right]
\right\} \\
&\leq &C\left( \frac{\bar{\varepsilon}}{2}\right) ^{-3/2}\sup_{(x,\tau )\in 
\mathcal{S}}\sup_{P\in \mathcal{P}}\left\{ \mathbf{E}\left[ ||q_{n,\tau
}(x)+\eta _{1}||^{3}\right] +\mathbf{E}\left[ |\tilde{U}_{n}|^{3}\right]
\right\} ,
\end{eqnarray*}%
where the last inequality uses (\ref{LB2}). As for the last expectation,
note that by Rosenthal's inequality, we have 
\begin{equation*}
\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\mathbf{E}\left[ |%
\tilde{U}_{n}|^{3}\right] \leq C
\end{equation*}%
for some $C>0$. We apply the first statement of Lemma B5 to conclude that%
\begin{equation*}
\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\mathbf{E}\left\Vert
W_{n,\tau }(x;\eta _{1})\right\Vert ^{3}\leq C\bar{\varepsilon}%
^{-3/2}h^{-d/2},
\end{equation*}%
for some $C>0$. For any vector $\mathbf{v}=[\mathbf{v}_{1}^{\top
},v_{2}]^{\top }\in \mathbf{R}^{J+1}$, we define%
\begin{equation*}
D_{n,\tau ,p}(\mathbf{v})\equiv \Lambda _{A,p}\left( \left[ \Sigma _{2n,\tau
,\bar{\varepsilon}}^{1/2}\mathbf{v}\right] _{1}\right) \left[ \Sigma
_{2n,\tau ,\bar{\varepsilon}}^{1/2}\mathbf{v}\right] _{2},
\end{equation*}%
where $[a]_{1}$ of a vector $a\in \mathbf{R}^{J+1}$ indicates the vector of
the first $J$ entries of $a$, and $[a]_{2}$ the last entry of $a.$ By
Theorem 1 of \citeasnoun{Sweeting:77}, we find that (with $\bar{\varepsilon}%
>0$ fixed)%
\begin{equation*}
\mathbf{E}\left[ D_{n,\tau ,p}\left( \frac{1}{\sqrt{n}}\sum_{i=1}^{n}W_{n,%
\tau }^{(i)}(x;\eta _{1})\right) \right] =\mathbf{E}\left[ D_{n,\tau
,p}\left( \mathbb{Z}_{J+1}\right) \right] +O(n^{-1/2}h^{-d/2}) = o(n^{d/2}),
\end{equation*}%
where $\mathbb{Z}_{J+1}\sim N(0,I_{J+1})$ and $W_{n,\tau }^{(i)}(x;\eta
_{1}) $'s are i.i.d. copies of $W_{n,\tau }(x;\eta _{1})$. The last equality follows because $
n^{-1/2}h^{-d/2}=o(h^{d/2})\ $(by the condition that $n^{-1/2}h^{-d-\nu_1
}\rightarrow 0$ for some small $\nu_1>0$ as $n\rightarrow \infty $) and $\mathbf{E}[D_{n,\tau ,p}\left( \mathbb{Z}_{J+1}\right) ]=0$. Since
\begin{equation*}
Cov\left( \Lambda _{A,p}\left( \sqrt{nh^{d}}\mathbf{z}_{N,\tau }(x;\eta
_{1})\right) ,U_{n}\right) =\mathbf{E}\left[ D_{n,\tau ,p}\left( \frac{1}{%
\sqrt{n}}\sum_{i=1}^{n}W_{n,\tau }^{(i)}(x;\eta _{1})\right) \right],
\end{equation*}
we conclude that 
\begin{equation*}
\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\left\vert Cov\left(
\Lambda _{A,p}\left( \sqrt{nh^{d}}\mathbf{z}_{N,\tau }(x;\eta _{1})\right)
,U_{n}\right) \right\vert =o(h^{d/2}).
\end{equation*}%
By applying the Dominated Convergence Theorem, we obtain Step 2.\medskip 
\newline
\textbf{Proof of Step 3: }First, we show that 
\begin{equation}
Var\left( S_{n}\right) =\sigma _{n}^{2}(\mathcal{C})+o(1),  \label{var}
\end{equation}%
where $o(1)$ is an asymptotically negligible term uniformly over $P\in 
\mathcal{P}$. Note that%
\begin{equation*}
Var\left( S_{n}\right) =\sum_{A\in \mathcal{N}_{J}}\sum_{A^{\prime }\in 
\mathcal{N}_{J}}\mu _{A}\mu _{A^{\prime }}Cov(\psi _{n,A},\psi _{n,A^{\prime
}}),
\end{equation*}%
where $\psi _{n,A}\equiv h^{-d/2}(\zeta _{N,A}-\mathbf{E}\zeta _{N,A})$. By
Lemma B6, we find that for $A,A^{\prime }\in \mathcal{N}_{J},$ 
\begin{equation*}
Cov(\psi _{n,A},\psi _{n,A^{\prime }})=\sigma _{n,A,A^{\prime
}}(B_{n,A}(c_{n};\mathcal{C}),B_{n,A^{\prime }}(c_{n};\mathcal{C}))+o(1),
\end{equation*}%
uniformly in $P\in \mathcal{P}$, yielding the desired result.

Combining Steps 1 and 2, we deduce that\textbf{\ }%
\begin{equation}
\sup_{P\in \mathcal{P}}\left\vert Cov(S_{n},U_{n})\right\vert \leq C\sqrt{%
\bar{\varepsilon}}+o(h^{d/2}).  \label{bd45}
\end{equation}%
Let $\bar{\sigma}_{1}^{2}\equiv \inf_{P\in \mathcal{P}}\sigma _{n}^{2}(%
\mathcal{C})$ and $\bar{\sigma}_{2}^{2}\equiv \inf_{P\in \mathcal{P}%
}(1-\alpha _{P})$. Note that for some $C_{1}>0$,%
\begin{equation}
\inf_{P\in \mathcal{P}}\bar{\sigma}_{1}^{2}\bar{\sigma}_{2}^{2}>C_{1},
\label{bd56}
\end{equation}%
by the condition of the lemma. A simple calculation gives us%
\begin{eqnarray}
\lambda _{\min }(C_{n}) &=&\frac{\bar{\sigma}_{1}^{2}+\bar{\sigma}_{2}^{2}}{2%
}-\frac{1}{2}\left( \sqrt{\left( \bar{\sigma}_{1}^{2}+\bar{\sigma}%
_{2}^{2}\right) ^{2}-4\left\{ \bar{\sigma}_{1}^{2}\bar{\sigma}%
_{2}^{2}-Cov(S_{n},U_{n})^{2}\right\} }\right)  \label{dev67} \\
&\geq &\frac{1}{2}\left\{ \sqrt{\left( \bar{\sigma}_{1}^{2}+\bar{\sigma}%
_{2}^{2}\right) ^{2}}-\left( \sqrt{\left( \bar{\sigma}_{1}^{2}+\bar{\sigma}%
_{2}^{2}\right) ^{2}-4\bar{\sigma}_{1}^{2}\bar{\sigma}_{2}^{2}}\right)
\right\} -\left\vert Cov(S_{n},U_{n})\right\vert  \notag \\
&\geq &\bar{\sigma}_{1}^{2}\bar{\sigma}_{2}^{2}-\left\vert
Cov(S_{n},U_{n})\right\vert \geq C_{1}-C\sqrt{\bar{\varepsilon}}+o(h^{d/2}),
\notag
\end{eqnarray}%
where the last inequality follows by (\ref{bd45}) and (\ref{bd56}). Taking $%
\bar{\varepsilon}$ small enough, we obtain the desired result.\medskip 
\newline
\textbf{Proof of Step 4:} Suppose that\textit{\ }liminf$_{n\rightarrow
\infty }\inf_{P\in \mathcal{P}}\sigma _{n}^{2}(\mathcal{C})>0$. Let $\kappa $
be the diameter of the compact set $\mathcal{K}_{0}$ introduced in
Assumption A2. Let $\mathcal{C}$ be given as in the lemma. Let $\mathbb{Z}%
^{d}$ be the set of $d$-tuples of integers, and let $\{R_{n,\mathbf{i}}:%
\mathbf{i}\in \mathbb{Z}^{d}\}$ be the collection of rectangles in $\mathbf{R%
}^{d}$ such that $R_{n,\mathbf{i}}=[a_{n,\mathbf{i}_{1}},b_{n,\mathbf{i}%
_{1}}]\times ... \times \lbrack a_{n,\mathbf{i}%
_{d}},b_{n,\mathbf{i}_{d}}]$, where $\mathbf{i}_{j}$ is the $j$-th entry of $%
\mathbf{i}$, and $h\kappa \leq b_{n,\mathbf{i}_{j}}-a_{n,\mathbf{i}_{j}}\leq
2h\kappa $, for all $j=1,...,d$, and two different rectangles 
$R_{n,\mathbf{i}}$ and $R_{n,\mathbf{j}}$ do not have intersection with
nonempty interior, and the union of the rectangles $R_{n,\mathbf{i}}$, $%
\mathbf{i}\in \mathbb{Z}_{n}^{d}$, cover $\mathcal{X}$ from some
sufficiently large $n$. Here, $\mathbb{Z}_{n}^{d}$ is the set of $d$%
-tuples of integers whose absolute values are less than or equal to $n$.

We let%
\begin{eqnarray*}
B_{n,A,x}(c_{n}) &\equiv &\left\{ \tau \in \mathcal{T}:(x,\tau )\in
B_{A}(c_{n})\right\} , \\
B_{n,\mathbf{i}} &\equiv &R_{n,\mathbf{i}}\cap \mathcal{C},
\end{eqnarray*}%
and $\mathcal{I}_{n}\equiv \{\mathbf{i}\in \mathbb{Z}_{n}^{d}:B_{n,\mathbf{i}%
}\neq \varnothing \}$. Then $B_{n,\mathbf{i}}$ has Lebesgue measure $m(B_{n,%
\mathbf{i}})$ bounded by $C_{1}h^{d}$ and the cardinality of the set $%
\mathcal{I}_{n}$ is bounded by $C_{2}h^{-d}$ for some positive constants $%
C_{1}$ and $C_{2}$. Now let us define%
\begin{equation*}
\Delta _{n,A,\mathbf{i}}\equiv h^{-d/2}\int_{B_{n,\mathbf{i}%
}}\int_{B_{n,A,x}(c_{n})}\left\{ \Lambda _{A,p}(\sqrt{nh^{d}}\mathbf{z}%
_{N,\tau }(x))-\mathbf{E}\left[ \Lambda _{A,p}(\sqrt{nh^{d}}\mathbf{z}%
_{N,\tau }(x))\right] \right\} d\tau dx.
\end{equation*}%
And also define $B_{n,A,\mathbf{i}}(c_{n})\equiv (B_{n,\mathbf{i}}\times 
\mathcal{T})\cap B_{n,A}(c_{n}),$%
\begin{eqnarray*}
\alpha _{n,\mathbf{i}} &\equiv &\frac{\sum_{A\in \mathcal{N}_{J}}\mu
_{A}\Delta _{n,A,\mathbf{i}}}{\sigma _{n}(\mathcal{C})}\mathbf{\ }\text{and}
\\
u_{n,\mathbf{i}} &\equiv &\frac{1}{\sqrt{n}}\left\{ \sum_{i=1}^{N}1\left\{
X_{i}\in B_{n,\mathbf{i}}\right\} -nP\{X_{i}\in B_{n,\mathbf{i}}\}\right\} .
\end{eqnarray*}%
Then, we can write%
\begin{equation*}
\frac{S_{n}}{\sigma _{n}(\mathcal{C})}=\sum_{\mathbf{i}\in \mathcal{I}%
_{n}}\alpha _{n,\mathbf{i}}\text{ and }U_{n}=\sum_{\mathbf{i}\in \mathcal{I}%
_{n}}u_{n,\mathbf{i}}.
\end{equation*}%
By the definition of $\mathcal{K}_{0}$ in Assumption A2, by the definition
of $R_{n,\mathbf{i}}$ and by the properties of Poisson processes, one can
see that the array $\{(\alpha _{n,\mathbf{i}},u_{n,\mathbf{i}})\}_{\mathbf{i}%
\in \mathcal{I}_{n}}$ is an array of $1$-dependent random field. (See %
\citeasnoun{Mason/Polonik:09} for details.) For any $q_{1},q_{2}\in \mathbf{R%
}$, let $y_{n,\mathbf{i}}\equiv q_{1}\alpha _{n,\mathbf{i}}+q_{2}u_{n,%
\mathbf{i}}$. The focus is on the convergence in distribution of $\sum_{%
\mathbf{i}\in \mathcal{I}_{n}}y_{n,\mathbf{i}}$ uniform over $P\in \mathcal{P%
}$. Without loss of generality, we choose $q_{1},q_{2}\in \mathbf{R}%
\backslash \{0\}$. Define%
\begin{equation*}
Var_{P}\left( \sum_{\mathbf{i}\in \mathcal{I}_{n}}y_{n,\mathbf{i}}\right)
=q_{1}^{2}+q_{2}^{2}(1-\alpha _{P})+2q_{1}q_{2}c_{n,P},
\end{equation*}%
uniformly over $P\in \mathcal{P}$, where $c_{n,P}=Cov(S_{n},U_{n})$. On the
other hand, using Lemma B4 and following the proof of Lemma A8 of %
\citeasnoun{LSW}, we deduce that%
\begin{equation}
\sup_{P\in \mathcal{P}}\sum_{\mathbf{i}\in \mathcal{I}_{n}}\mathbf{E}|y_{n,%
\mathbf{i}}|^{r}=o(1)  \label{conv_y}
\end{equation}%
as $n\rightarrow \infty $, for any $r\in (2,(2p+2)/p]$. By Theorem 1 of %
\citeasnoun{Shergin:93}, we have%
\begin{eqnarray*}
&&\sup_{P\in \mathcal{P}}\sup_{t\in \mathbf{R}}\left\vert P\left\{ \frac{1}{%
\sqrt{q_{1}^{2}+q_{2}^{2}(1-\alpha _{P})+2q_{1}q_{2}c_{n,P}}}\sum_{\mathbf{i}%
\in \mathcal{I}_{n}}y_{n,\mathbf{i}}\leq t\right\} -\Phi \left( t\right)
\right\vert \\
&\leq &\sup_{P\in \mathcal{P}}\frac{C}{\left\{ q_{1}^{2}+q_{2}^{2}(1-\alpha
_{P})+2q_{1}q_{2}c_{n,P}\right\} ^{r/2}}\left\{ \sum_{\mathbf{i}\in \mathcal{%
I}_{n}}\mathbf{E}|y_{n,\mathbf{i}}|^{r}\right\} ^{1/2}=o(1),
\end{eqnarray*}%
for some $C>0$, by (\ref{conv_y}). Therefore, by Lemma B2(i), we have for
each $t\in \mathbf{R},$ and each $q\in \mathbf{R}^{2}\backslash \{0\}$, as $%
n\rightarrow \infty ,$%
\begin{equation*}
\sup_{P\in \mathcal{P}}\left\vert \mathbf{E}\left[ \exp \left( it\frac{%
q^{\top }H_{n}}{\sqrt{q^{\top }C_{n}q}}\right) \right] -\exp \left( -\frac{%
t^{2}}{2}\right) \right\vert \rightarrow 0.
\end{equation*}%
Thus by Lemma B2(ii), for each $t\in \mathbf{R}^{2},$ we have%
\begin{equation*}
\sup_{P\in \mathcal{P}}\left\vert P\left\{ C_{n}^{-1/2}H_{n}\leq t\right\}
-P\left\{ \mathbb{Z}\leq t\right\} \right\vert \rightarrow 0.
\end{equation*}%
Since the limit distribution of $C_{n}^{-1/2}H_{n}$ is continuous, the
convergence above is uniform in $t\in \mathbf{R}^{2}$.\medskip

\noindent (ii) We fix $P\in \mathcal{P}$ such that limsup$_{n\rightarrow
\infty }\sigma _{n}^{2}(\mathcal{C})=0.$ Then by (\ref{var}) above, 
\begin{equation*}
Var\left( S_{n}\right) =\sigma _{n}^{2}(\mathcal{C})+o(1)=o(1).
\end{equation*}%
Hence, we find that $S_{n}=o_{P}(1)$. The desired result follows by applying
Theorem 1 of \citeasnoun{Shergin:93} to the sum $U_{n}=\sum_{\mathbf{i}\in 
\mathcal{I}_{n}}u_{n,\mathbf{i}}$, and then applying Lemma B2(ii).
\end{proof}

\begin{LemmaC}
\label{lem-c9} \textit{Let }$\mathcal{C}$\textit{\ be the Borel set in Lemma
B8}.

\noindent \textit{(i) Suppose that the conditions of Lemma B8(i) are
satisfied. Then as }$n\rightarrow
\infty ,$%
\begin{equation*}
\sup_{P\in \mathcal{P}}\sup_{t\in \mathbf{R}}\left\vert P\left\{ \frac{%
h^{-d/2}\sum_{A\in \mathcal{N}_{J}}\mu _{A}\left\{ \zeta _{n,A}-\mathbf{E}%
\zeta _{N,A}\right\} }{\sigma _{n}(\mathcal{C})}\leq t\right\} -\Phi
(t)\right\vert \rightarrow 0.
\end{equation*}%
\noindent \textit{(ii) Suppose that the conditions of Lemma B8(ii) are
satisfied. Then as }$n\rightarrow \infty ,$%
\begin{equation*}
h^{-d/2}\sum_{A\in \mathcal{N}_{J}}\mu _{A}\left\{ \zeta _{n,A}-\mathbf{E}%
\zeta _{N,A}\right\} \overset{p}{\rightarrow }0.
\end{equation*}
\end{LemmaC}

Note that in both statements, the location normalization has $\mathbf{E}%
\zeta _{N,A}$ instead of $\mathbf{E}\zeta _{n,A}$.

\begin{proof}[Proof of Lemma B\protect\ref{lem-c9}]
(i) The conditional distribution of $S_{n}/\sigma _{n}(\mathcal{C})$ given $%
N=n$ is equal to that of 
\begin{equation*}
\frac{\sum_{A\in \mathcal{N}_{J}}\mu _{A}\int_{B_{n,A}(c_{n};\mathcal{C}%
)\cap \mathcal{C}}\left\{ \Lambda _{A,p}(\sqrt{nh^{d}}\mathbf{z}_{n,\tau
}(x))-\mathbf{E}\Lambda _{A,p}(\sqrt{nh^{d}}\mathbf{z}_{N,\tau }(x))\right\}
dQ(x,\tau )}{h^{d/2}\sigma _{n}(\mathcal{C})}.
\end{equation*}%
Using Lemmas B3(i) and B8(i), we find that%
\begin{equation*}
\frac{h^{-d/2}\sum_{A\in \mathcal{N}_{J}}\mu _{A}\left\{ \zeta _{n,A}-%
\mathbf{E}\zeta _{N,A}\right\} }{\sigma _{n}(\mathcal{C})}\overset{d}{%
\rightarrow }N(0,1).
\end{equation*}%
Since the limit distribution $N(0,1)$ is continuous and the convergence is
uniform in $P\in \mathcal{P}$, we obtain the desired result.

\noindent (ii) Similarly as before, the result follows from Lemmas B3(ii),
B2(ii), and B8(ii).
\end{proof}

\section{Proofs of Auxiliary Results for Lemmas A2(ii), Lemma A4(ii), and
Theorem 1}

\label{appendix:D}

The auxiliary results in this section are mostly bootstrap versions of the
results in Appendix B. To facilitate comparison, we name the first lemma to
be Lemma C3, which is used to control the discrepancy between the sample
version of the scale normalizer $\sigma _{n}$, and its population version.
Then we proceed to prove Lemmas C4-C9 which run in parallel with Lemmas
B4-B9 as their bootstrap counterparts. We finish this subsection with Lemmas
C10-C12 which are crucial for dealing with the bootstrap test statistic's
location normalization. More specifically, Lemmas C10 and C11 are auxiliary
moment bound results that are used for proving Lemma C12. Lemma C12
essentially delivers the result of Lemma A1 in Appendix A. This lemma is
used to deal with the discrepancy between the population location normalizer
and the sample location normalizer. Controlling this discrepancy to the rate 
$o_{P}(h^{d/2})$ is crucial for our purpose, because the bootstrap test
statistic that is proposed here does not involve the sample version of the location normalizer $
a_{n}$ for the sake of computational expediency. Lemmas C10 and C11 provide necessary
moment bounds to achieve this convergence rate.

Let the random variables $N$ and $N_{1}$ represent Poisson random variables with
mean $n$ and $1$ respectively. These random variables are taken to be independent of $%
( (Y_i^*, X_i^*)_{i=1}^{\infty},(Y_i,X_i)_{i=1}^{\infty })$. Let $\eta _{1}$ and $\eta _{2}$
be centered normal random vectors that are independent of each other and
independent of 
\begin{equation*}
\left( (Y_i^*,X_i^*)_{i=1}^{\infty },(Y_i,X_i)_{i=1}^{\infty },N,N_{1}\right) .
\end{equation*}%
We will specify their covariance matrices in the proofs below. Throughout
the proofs, the bootstrap distribution $P^{\ast }$ and expectations $\mathbf{%
E}^{\ast }$ are viewed as the distribution of 
\begin{equation*}
\left( (Y_{i}^{\ast },X_{i}^{\ast })_{i=1}^{n},N,N_{1},\eta _{1},\eta
_{2}\right) ,
\end{equation*}%
conditional on $(Y_{i},X_{i})_{i=1}^{n}$.

Define 
\begin{eqnarray*}
\tilde{\rho}_{n,\tau _{1},\tau _{2},j,k}(x,u) &\equiv &\frac{1}{h^{d}}%
\mathbf{E}^{\ast }\left[ \beta _{n,x,\tau _{1},j}\left( Y_{ij}^{\ast },\frac{%
X_{i}^{\ast }-x}{h}\right) \beta _{n,x,\tau _{2},k}\left( Y_{ik}^{\ast },%
\frac{X_{i}^{\ast }-x}{h}+u\right) \right] \text{ and } \\
\tilde{k}_{n,\tau ,j,m}(x) &\equiv &\frac{1}{h^{d}}\mathbf{E}^{\ast }\left[
\left\vert \beta _{n,x,\tau ,j}\left( Y_{ij}^{\ast },\frac{X_{i}^{\ast }-x}{h%
}\right) \right\vert ^{m}\right] .
\end{eqnarray*}%
Note that $\tilde{\rho}_{n,\tau _{1},\tau _{2},j,k}(x,u)$ and $\tilde{k}%
_{n,\tau ,j,m}(x)$ are bootstrap versions of $\rho _{n,\tau _{1},\tau
_{2},j,k}(x,u)$ and $\tilde{k}_{n,\tau ,j,m}(x)$. The lemma below
establishes that the bootstrap version $\tilde{\rho}_{n,\tau _{1},\tau
_{2},j,k}(x,u)$ is consistent for $\rho _{n,\tau _{1},\tau _{2},j,k}(x,u)$.

\setcounter{LemmaD}{2}

\begin{LemmaD}
\label{lem-d3} \textit{Suppose that Assumption A\ref{assumption-A6}(i) holds
and that }$n^{-1/2}h^{-d/2}\rightarrow 0$\textit{, as }$n\rightarrow \infty $%
.\textit{\ Then for each }$\varepsilon \in (0,\varepsilon _{1})$\textit{,
with }$\varepsilon _{1}>0$\textit{\ as in Assumption A6(i), as }$%
n\rightarrow \infty $,%
\begin{equation*}
\sup_{(\tau _{1},\tau _{2})\in \mathcal{T}\times \mathcal{T}}\sup_{(x,u)\in (%
\mathcal{S}_{\tau _{1}}(\varepsilon )\cup \mathcal{S}_{\tau
_{2}}(\varepsilon ))\times \mathcal{U}}\sup_{P\in \mathcal{P}}\mathbf{E}%
\left( \left\vert \tilde{\rho}_{n,\tau _{1},\tau _{2},j,k}(x,u)-\rho
_{n,\tau _{1},\tau _{2},j,k}(x,u)\right\vert ^{2}\right) \rightarrow 0.
\end{equation*}
\end{LemmaD}

\begin{proof}[Proof of Lemma C\protect\ref{lem-d3}]
Define $\pi _{n,x,u,\tau _{1},\tau _{2}}(y,z)=\beta _{n,x,\tau
_{1},j}(y_{j},(z-x)/h)\beta _{n,x,\tau _{2},k}(y_{k},(z-x)/h+u)$ for $%
y=(y_{1},...,y_{J})^{\top }\in \mathbf{R}^{J}$, and write%
\begin{equation*}
\tilde{\rho}_{n,\tau _{1},\tau _{2},j,k}(x,u)-\rho _{n,\tau _{1},\tau
_{2},j,k}(x,u)=\frac{1}{nh^{d}}\sum_{i=1}^{n}\left\{ \pi _{n,x,u,\tau
_{1},\tau _{2}}(Y_{i},X_{i})-\mathbf{E}\left[ \pi _{n,x,u,\tau _{1},\tau
_{2}}(Y_{i},X_{i})\right] \right\} .
\end{equation*}%
First, we note that%
\begin{equation*}
\mathbf{E}\left( \frac{1}{\sqrt{n}}\sum_{i=1}^{n}\left\{ \pi _{n,x,u,\tau
_{1},\tau _{2}}(Y_{i},X_{i})-\mathbf{E}\left[ \pi _{n,x,u,\tau _{1},\tau
_{2}}(Y_{i},X_{i})\right] \right\} \right) ^{2}\leq \mathbf{E}\left[ \pi
_{n,x,u,\tau _{1},\tau _{2}}^{2}(Y_{i},X_{i})\right] .
\end{equation*}%
By change of variables and Assumption A\ref{assumption-A6}(i), we have $%
\mathbf{E}\left[ \pi _{n,x,u,\tau _{1},\tau _{2}}^{2}(Y_{i},X_{i})\right]
=O(h^{d})$ uniformly over $(\tau _{1},\tau _{2})\in \mathcal{T}\times 
\mathcal{T}$, $(x,u)\in (\mathcal{S}_{\tau _{1}}(\varepsilon )\cup \mathcal{S%
}_{\tau _{2}}(\varepsilon ))\times \mathcal{U}$ and over $P\in \mathcal{P}$.
Hence%
\begin{equation*}
\mathbf{E}\left( \left\vert \tilde{\rho}_{n,\tau _{1},\tau
_{2},j,k}(x,u)-\rho _{n,\tau _{1},\tau _{2},j,k}(x,u)\right\vert ^{2}\right)
=O\left( n^{-1}h^{-d}\right) ,
\end{equation*}%
uniformly over $(\tau _{1},\tau _{2})\in \mathcal{T}\times \mathcal{T}$, $%
(x,u)\in (\mathcal{S}_{\tau _{1}}(\varepsilon )\cup \mathcal{S}_{\tau
_{2}}(\varepsilon ))\times \mathcal{U}$ and over $P\in \mathcal{P}$. Since
we have assumed that$\ n^{-1/2}h^{-d/2}\rightarrow 0$ as $n\rightarrow
\infty $, we obtain the desired result.
\end{proof}

\begin{LemmaD}
\label{lem-d4} \textit{Suppose that Assumption A\ref{assumption-A6}(i) holds
and that for some }$C>0,$%
\begin{equation*}
\limsup_{n\rightarrow \infty }n^{-1/2}h^{-d/2}\leq C\text{.}
\end{equation*}%
\textit{\ Then for all }$m\in \lbrack 2,M]$\textit{\ and all }$\varepsilon
\in (0,\varepsilon _{1})$\textit{, with }$M>2$\textit{\ and }$\varepsilon
_{1}>0$\textit{\ being the constants that appear in Assumption A\ref%
{assumption-A6}(i)), there exists }$C_{1}\in (0,\infty )$\textit{\ that does
not depend on }$n$\textit{\ such that for each }$j\in \mathbb{N}_{J}$\textit{%
,}%
\begin{equation*}
\sup_{\tau \in \mathcal{T},x\in \mathcal{S}_{\tau }(\varepsilon )}\sup_{P\in 
\mathcal{P}}\mathbf{E}\left[ \tilde{k}_{n,\tau ,j,m}^{2}(x)\right] \leq
C_{1}.
\end{equation*}
\end{LemmaD}

\begin{proof}[Proof of Lemma C\protect\ref{lem-d4}]
Since $\mathbf{E}^{\ast }[|\beta _{n,x,\tau ,j}(Y_{ij}^{\ast },(X_{i}^{\ast
}-x)/h)|^{m}]=\frac{1}{n}\sum_{i=1}^{n}|\beta _{n,x,\tau
,j}(Y_{ij},(X_{i}-x)/h)|^{m}$, we find that%
\begin{equation*}
\tilde{k}_{n,\tau ,j,m}^{2}(x)\leq 2k_{n,\tau ,j,m}^{2}(x)+2e_{n,\tau
,j,m}^{2}(x),
\end{equation*}%
where%
\begin{equation*}
e_{n,\tau ,j,m}(x)\equiv \left\vert \frac{1}{nh^{d}}\sum_{i=1}^{n}\left\vert
\beta _{n,x,\tau ,j}\left( Y_{ij},\frac{X_{i}-x}{h}\right) \right\vert ^{m}-%
\frac{1}{h^{d}}\mathbf{E}\left( \left\vert \beta _{n,x,\tau ,j}\left( Y_{ij},%
\frac{X_{i}-x}{h}\right) \right\vert ^{m}\right) \right\vert .
\end{equation*}%
Similarly as in the proof of Lemma C3, we note that%
\begin{eqnarray*}
&&\sup_{\tau \in \mathcal{T},x\in \mathcal{S}_{\tau }(\varepsilon
)}\sup_{P\in \mathcal{P}}\mathbf{E}\left[ \left\vert e_{n,\tau
,j,m}^{2}(x)\right\vert \right] \\
&\leq &\sup_{\tau \in \mathcal{T},x\in \mathcal{S}_{\tau }(\varepsilon
)}\sup_{P\in \mathcal{P}}\frac{1}{nh^{2d}}\mathbf{E}\left[ \left\vert \beta
_{n,x,\tau ,j}\left( Y_{ij},\frac{X_{i}-x}{h}\right) \right\vert ^{2m}\right]
=O(n^{-1}h^{-d})=o(1),\text{ as }n\rightarrow \infty \text{.}
\end{eqnarray*}%
Hence the desired statement follows from Lemma B4.
\end{proof}

Let%
\begin{eqnarray*}
\mathbf{z}_{n,\tau }^{\ast }(x) &\equiv &\frac{1}{nh^{d}}\sum_{i=1}^{n}\beta
_{n,x,\tau }\left( Y_{i}^{\ast },\frac{X_{i}^{\ast }-x}{h}\right) -\frac{1}{%
h^{d}}\mathbf{E}^{\ast }\left[ \beta _{n,x,\tau }\left( Y_{i}^{\ast },\frac{%
X_{i}^{\ast }-x}{h}\right) \right] \text{, and} \\
\mathbf{z}_{N,\tau }^{\ast }(x) &\equiv &\frac{1}{nh^{d}}\sum_{i=1}^{N}\beta
_{n,x,\tau }\left( Y_{i}^{\ast },\frac{X_{i}^{\ast }-x}{h}\right) -\frac{1}{%
h^{d}}\mathbf{E}^{\ast }\left[ \beta _{n,x,\tau }\left( Y_{i}^{\ast },\frac{%
X_{i}^{\ast }-x}{h}\right) \right] .
\end{eqnarray*}%
We also let%
\begin{eqnarray*}
q_{n,\tau }^{\ast }(x) &\equiv &\frac{1}{\sqrt{h^{d}}}\sum_{i\leq
N_{1}}\left\{ \beta _{n,x,\tau }(Y_{i}^{\ast },(X_{i}^{\ast }-x)/h)-\mathbf{E%
}^{\ast }\beta _{n,x,\tau }(Y_{i}^{\ast },(X_{i}^{\ast }-x)/h)\right\} \ 
\text{and} \\
\bar{q}_{n,\tau }^{\ast }(x) &\equiv &\frac{1}{\sqrt{h^{d}}}\left\{ \beta
_{n,x,\tau }(Y_{i}^{\ast },(X_{i}^{\ast }-x)/h)-\mathbf{E}^{\ast }\beta
_{n,x,\tau }(Y_{i}^{\ast },(X_{i}^{\ast }-x)/h)\right\} .
\end{eqnarray*}

\begin{LemmaD}
\label{lem-d5} \textit{Suppose that Assumption A\ref{assumption-A6}(i) holds
and that for some }$C>0,$%
\begin{equation*}
\text{limsup}_{n\rightarrow \infty }n^{-1/2}h^{-d/2}\leq C\text{.}
\end{equation*}%
\textit{Then for any }$m\in \lbrack 2,M]$ \textit{(with }$M$ \textit{being
the constant }$M$\textit{\ in Assumption A6(i)), }%
\begin{eqnarray}
\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\sqrt{\mathbf{E}\left[
\left( \mathbf{E}^{\ast }\left[ ||q_{n,\tau }^{\ast }(x)||^{m}\right]
\right) ^{2}\right] } &\leq &\bar{C}_{1}h^{d(1-(m/2))},\text{\textit{\ and}}
\label{bd11} \\
\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\sqrt{\mathbf{E}\left[
\left( \mathbf{E}^{\ast }\left[ ||\bar{q}_{n,\tau }^{\ast }(x)||^{m}\right]
\right) ^{2}\right] } &\leq &\bar{C}_{2}h^{d(1-(m/2))},  \notag
\end{eqnarray}%
\textit{where }$\bar{C}_{1},\bar{C}_{2}>0$\textit{\ are constants that
depend only on }$m.$ \textit{If furthermore, }%
\begin{equation*}
\limsup_{n\rightarrow \infty }n^{-(m/2)+1}h^{d(1-(m/2))}<C,
\end{equation*}%
\textit{for some constant} $C>0$, \textit{then} 
\begin{eqnarray}
\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\mathbf{E}\left[ 
\mathbf{E}^{\ast }\left[ ||\sqrt{nh^{d}}\mathbf{z}_{N,\tau }^{\ast }(x)||^{m}%
\right] \right] &\leq &\left( \frac{15m}{\log m}\right) ^{m}\max \left\{ 
\bar{C}_{1},2\bar{C}_{1}C\right\} \text{,\textit{\ and}}  \label{bds22} \\
\sup_{(x,\tau )\in \mathcal{X}^{\varepsilon /2}\times \mathcal{T}}\sup_{P\in 
\mathcal{P}}\mathbf{E}\left[ \mathbf{E}^{\ast }\left[ ||\sqrt{nh^{d}}\mathbf{%
z}_{n,\tau }^{\ast }(x)||^{m}\right] \right] &\leq &\left( \frac{15m}{\log m}%
\right) ^{m}\max \left\{ \bar{C}_{2},2\bar{C}_{2}C\right\} ,  \notag
\end{eqnarray}%
\textit{where} $\bar{C}_{1},\bar{C}_{2}>0$\textit{\ are the constants that
appear in (\ref{bd11}).}
\end{LemmaD}

\begin{proof}[Proof of Lemma C\protect\ref{lem-d5}]
Let $q_{n,\tau ,j}^{\ast }(x)$ be the $j$-th entry of $q_{n,\tau }^{\ast
}(x) $. For the first statement of the lemma, it suffices to observe that
for each $\varepsilon \in (0,\varepsilon _{1})$, there exist $C_{1}>0$ and $%
\bar{C}_{1}>0$ such that%
\begin{eqnarray*}
&& \sup_{\tau \in \mathcal{T},x\in \mathcal{S}_{\tau }(\varepsilon )}\mathbf{E}%
\left[ \left( \mathbf{E}^{\ast }\left[ |q_{n,\tau ,j}^{\ast }(x)|^{m}\right]
\right) ^{2}\right] \\
&\le& \frac{C_{1}h^{2d}\sum_{j=1}^{J}\sup_{\tau \in 
\mathcal{T},x\in \mathcal{S}_{\tau }(\varepsilon )}\sup_{P\in \mathcal{P}}%
\mathbf{E}\left[ \tilde{k}_{n,\tau ,j,m}^{2}(x)\right] }{h^{dm}}\leq \bar{C}%
_{1}h^{2d(1-(m/2))},
\end{eqnarray*}%
where the last inequality uses Lemma C4. The second inequality in (\ref{bd11}%
) follows similarly.

Let us consider (\ref{bds22}). Let $z_{N,\tau ,j}^{\ast }(x)$ be the $j$-th
entry of $\mathbf{z}_{N,\tau }^{\ast }(x)$. Then using Rosenthal's
inequality (e.g. (2.3) of \citeasnoun{GMZ}), for some constant $C_{1}>0,$%
\begin{eqnarray*}
&&\sup_{\tau \in \mathcal{T},x\in \mathcal{S}_{\tau }(\varepsilon
)}\sup_{P\in \mathcal{P}}\mathbf{E}\left[ \mathbf{E}^{\ast }[|\sqrt{nh^{d}}%
z_{N,\tau ,j}^{\ast }(x)|^{m}]\right] \\
&\leq &\left( \frac{15m}{\log m}\right) ^{2m}\sup_{\tau \in \mathcal{T},x\in 
\mathcal{S}_{\tau }(\varepsilon )}\sup_{P\in \mathcal{P}}\left\{ \left( 
\mathbf{E}\left[ \mathbf{E}^{\ast }\left( q_{n,\tau ,j}^{\ast 2}(x)\right) %
\right] \right) ^{m/2}+\mathbf{E}\left[ n^{-(m/2)+1}\mathbf{E}^{\ast
}|q_{n,\tau ,j}^{\ast }(x)|^{m}\right] \right\} .
\end{eqnarray*}%
The first expectation is bounded by $\bar{C}_{1}$ by (\ref{bd11}).

The second expectation is bounded by $\bar{C}_{1}n^{-(m/2)+1}h^{d(1-(m/2))}$%
. This gives the first bound in (\ref{bds22}). The second bound in (\ref%
{bds22}) can be obtained similarly.
\end{proof}

For any Borel sets $B,B^{\prime }\subset \mathcal{S}$ and $A,A^{\prime
}\subset \mathbb{N}_{J},$ let%
\begin{equation*}
\tilde{\sigma}_{n,A,A^{\prime }}^{R}(B,B^{\prime })\equiv \int_{\mathcal{T}%
}\int_{\mathcal{T}}\int_{B_{\tau _{2}}^{\prime }}\int_{B_{\tau
_{1}}}C_{n,\tau _{1},\tau _{2},A,A^{\prime }}^{\ast }(x,v)dxdvd\tau
_{1}d\tau _{2},
\end{equation*}%
where $B_{\tau }\equiv \{x\in \mathcal{X}:(x,\tau )\in B\},$ 
\begin{equation}
C_{n,\tau _{1},\tau _{2},A,A^{\prime }}^{\ast }(x,v)\equiv h^{-d}Cov^{\ast
}\left( \Lambda _{A,p}(\sqrt{nh^{d}}\mathbf{z}_{N,\tau _{1}}^{\ast
}(x)),\Lambda _{A^{\prime },p}(\sqrt{nh^{d}}\mathbf{z}_{N,\tau _{2}}^{\ast
}(v))\right) ,  \label{C*}
\end{equation}%
and $Cov^{\ast }$ represents covariance under $P^{\ast }$. We also define 
\begin{equation}
\tilde{\sigma}_{n,A}^{R}(B)\equiv \tilde{\sigma}_{n,A,A}^{R}(B,B),
\label{sigma_tilde}
\end{equation}%
for brevity. Also, let $\Sigma _{n,\tau _{1},\tau _{2}}^{\ast }(x,u)$ be the $%
J\times J$ matrix whose $(j,k)$-th entry is given by $\tilde{\rho}_{n,\tau
_{1},\tau _{2},j,k}(x,u)$. Fix $\bar{\varepsilon}>0$ and define%
\begin{equation*}
\tilde{\Sigma}_{n,\tau _{1},\tau _{2},\bar{\varepsilon}}^{\ast }(x,u)\equiv %
\left[ 
\begin{array}{c}
\Sigma _{n,\tau _{1},\tau _{1}}^{\ast }(x,0)+\bar{\varepsilon}I_{J} \\ 
\Sigma _{n,\tau _{1},\tau _{2}}^{\ast }(x,u)%
\end{array}%
\begin{array}{c}
\Sigma _{n,\tau _{1},\tau _{2}}^{\ast }(x,u) \\ 
\Sigma _{n,\tau _{2},\tau _{2}}^{\ast }(x,0)+\bar{\varepsilon}I_{J}%
\end{array}%
\right] .
\end{equation*}%
We also define%
\begin{equation*}
\mathbf{\xi }_{N,\tau _{1},\tau _{2}}^{\ast }(x,u;\eta _{1},\eta _{2})\equiv 
\sqrt{nh^{d}}\Sigma _{n,\tau _{1},\tau _{2},\bar{\varepsilon}}^{\ast
-1/2}(x,u)\left[ 
\begin{array}{c}
\mathbf{z}_{N,\tau _{1}}^{\ast }(x;\eta _{1}) \\ 
\mathbf{z}_{N,\tau _{2}}^{\ast }(x+uh;\eta _{2})%
\end{array}%
\right] ,
\end{equation*}%
where $\eta _{1}\in \mathbf{R}^{J}$ and $\eta _{2}\in \mathbf{R}^{J}$ are
random vectors that are independent, and independent of $((Y_{i}^{\ast
},X_{i}^{\ast })_{i=1}^{\infty },(Y_{i},X_{i})_{i=1}^{\infty },N,N_{1})$,
each following $N(0,\bar{\varepsilon}I_{J})$, and define $\mathbf{z}_{N,\tau
}^{\ast }(x;\eta _{1})\equiv \mathbf{z}_{N,\tau }^{\ast }(x)+\eta _{1}/\sqrt{%
nh^{d}}.$

\begin{LemmaD}
\label{lem-d6} \textit{Suppose that\ Assumption A\ref{assumption-A6}(i)
holds and that }$nh^{d}\rightarrow \infty $\textit{, and }%
\begin{equation*}
\limsup_{n\rightarrow \infty }n^{-(m/2)+1}h^{d(1-(m/2))}<C,
\end{equation*}%
\textit{for some }$C>0$\textit{\ and some }$m\in \lbrack 2(p+1),M]$.

\textit{Then for any sequences of Borel sets }$B_{n},B_{n}^{\prime }\subset 
\mathcal{S}$\textit{\ and for any }$A,A^{\prime }\subset \mathbb{N}_{J},$%
\begin{equation*}
\sup_{P\in \mathcal{P}}\mathbf{E}\left( \left\vert \tilde{\sigma}%
_{n,A,A^{\prime }}^{R}(B_{n},B_{n}^{\prime }-\sigma
_{n,A,A^{\prime }}(B_{n},B_{n}^{\prime })\right\vert \right) \rightarrow
0,
\end{equation*}%
\textit{where }$\sigma _{n,A,A^{\prime }}(B_{n},B_{n}^{\prime })$\textit{%
\ is as defined in (\ref{s2})}.
\end{LemmaD}

\begin{proof}[Proof of Lemma C\protect\ref{lem-d6}]
The proof is very similar to that of Lemma B6. For brevity, we sketch the
proof here. Define for $\bar{\varepsilon}>0,$%
\begin{eqnarray*}
\tilde{\sigma}_{n,A,A^{\prime },\bar{\varepsilon}}^{R}(B_{n},B_{n}^{\prime
}) &\equiv &\int_{\mathcal{T}}\int_{\mathcal{T}}\int_{B_{n,\tau _{1}}}\int_{%
\mathcal{U}}\tilde{g}_{1n,\tau _{1},\tau _{2},\bar{\varepsilon}}(x,u)w_{\tau
_{1},B_{n}}(x)w_{\tau _{2},B_{n}^{\prime }}(x+uh)dudxd\tau _{1}d\tau _{2}, \\
\tilde{\tau}_{n,A,A^{\prime },\bar{\varepsilon}}(B_{n},B_{n}^{\prime })
&\equiv &\int_{\mathcal{T}}\int_{\mathcal{T}}\int_{B_{n,\tau _{1}}}\int_{%
\mathcal{U}}\tilde{g}_{2n,\tau _{1},\tau _{2},\bar{\varepsilon}}(x,u)w_{\tau
_{1},B_{n}}(x)w_{\tau _{2},B_{n}^{\prime }}(x+uh)dudxd\tau _{1}d\tau _{2},
\end{eqnarray*}%
where%
\begin{eqnarray*}
\tilde{g}_{1n,\tau _{1},\tau _{2},\bar{\varepsilon}}(x,u) &\equiv
&h^{-d}Cov^{\ast }(\Lambda _{A,p}(\sqrt{nh^{d}}\mathbf{z}_{N,\tau
_{1}}^{\ast }(x;\eta _{1})),\Lambda _{A^{\prime },p}(\sqrt{nh^{d}}\mathbf{z}%
_{N,\tau _{2}}^{\ast }(x+uh;\eta _{2}))),\text{ and} \\
\tilde{g}_{2n,\tau _{1},\tau _{2},\bar{\varepsilon}}(x,u) &\equiv &Cov^{\ast
}(\Lambda _{A,p}(\mathbb{\tilde{Z}}_{n,\tau _{1},\tau _{2},\bar{\varepsilon}%
}(x)),\Lambda _{A^{\prime },p}(\mathbb{\tilde{Z}}_{n,\tau _{1},\tau _{2},%
\bar{\varepsilon}}(x+uh))),
\end{eqnarray*}%
and $[\mathbb{\tilde{Z}}_{n,\tau _{1},\tau _{2},\bar{\varepsilon}}^{\top
}(x),\mathbb{\tilde{Z}}_{n,\tau _{1},\tau _{2},\bar{\varepsilon}}^{\top
}(z)]^{\top }$ is a centered normal $\mathbf{R}^{2J}$-valued random vector
with the same covariance matrix as the covariance matrix of $[\sqrt{nh^{d}}%
\mathbf{z}_{N,\tau _{1}}^{\ast \top }(x;\eta _{1}),\sqrt{nh^{d}}\mathbf{z}%
_{N,\tau _{2}}^{\ast \top }(z;\eta _{2})]^{\top }$ under the product measure
of the bootstrap distribution $P^{\ast }$ and the distribution of $(\eta
_{1}^{\top },\eta _{2}^{\top })^{\top }$. As in the proof of Lemma B6, it
suffices for the lemma to show the following two statements.

\noindent (\textit{Step 1}): As $n\rightarrow \infty ,$%
\begin{eqnarray*}
\sup_{P\in \mathcal{P}}\mathbf{E}\left( \left\vert \tilde{\sigma}%
_{n,A,A^{\prime },\bar{\varepsilon}}^{R}(B_{n},B_{n}^{\prime })-\tilde{\tau}%
_{n,A,A^{\prime },\bar{\varepsilon}}(B_{n},B_{n}^{\prime })\right\vert
\right) &\rightarrow &0,\text{ and} \\
\sup_{P\in \mathcal{P}}\mathbf{E}\left( \left\vert \tilde{\tau}%
_{n,A,A^{\prime },\bar{\varepsilon}}(B_{n},B_{n}^{\prime })-\sigma
_{n,A,A^{\prime },\bar{\varepsilon}}(B_{n},B_{n}^{\prime })\right\vert
\right) &\rightarrow &0.
\end{eqnarray*}

\noindent (\textit{Step 2}): For some $C>0$ that does not depend on $\bar{%
\varepsilon}\ $or $n,$ 
\begin{equation*}
\sup_{P\in \mathcal{P}}|\tilde{\sigma}_{n,A,A^{\prime },\bar{\varepsilon}%
}^{R}(B_{n},B_{n}^{\prime })-\tilde{\sigma}_{n,A,A^{\prime
}}^{R}(B_{n},B_{n}^{\prime })|\leq C\sqrt{\bar{\varepsilon}}.
\end{equation*}%
Then the desired result follows by sending $n\rightarrow \infty $ and $\bar{%
\varepsilon}\downarrow 0$, while chaining Steps 1 and 2 and the second
convergence in Step 2 in the proof of Lemma B6.

We first focus on the first statement of Step 1. For any vector $\mathbf{v}%
=[\mathbf{v}_{1}^{\top },\mathbf{v}_{2}^{\top }]^{\top }\in \mathbf{R}^{2J}$%
, we define%
\begin{equation}
\tilde{C}_{n,p}(\mathbf{v})\equiv \Lambda _{A,p}\left( \left[ \tilde{\Sigma}%
_{n,\tau _{1},\tau _{2},\bar{\varepsilon}}^{\ast 1/2}(x,u)\mathbf{v}\right]
_{1}\right) \Lambda _{A^{\prime },p}\left( \left[ \tilde{\Sigma}_{n,\tau
_{1},\tau _{2},\bar{\varepsilon}}^{\ast 1/2}(x,u)\mathbf{v}\right]
_{2}\right) ,  \label{lambda_b}
\end{equation}%
where $[a]_{1}$ of a vector $a\in \mathbf{R}^{2J}$ indicates the vector of
the first $J$ entries of $a$, and $[a]_{2}$ the vector of the remaining $J$
entries of $a.$ Also, similarly as in (\ref{lb2}),%
\begin{equation}
\lambda _{\min }\left( \tilde{\Sigma}_{n,\tau _{1},\tau _{2},\bar{\varepsilon%
}}^{\ast }(x,u)\right) \geq \bar{\varepsilon}.  \label{lambda_min_b}
\end{equation}

Let $\bar{q}_{n,\tau }^{\ast }(x;\eta _{1})$ be the column vector of entries 
$\bar{q}_{n,\tau ,j}^{\ast }(x;\eta _{1j})$ with $j$ running in the set $%
\mathbb{N}_{J}$, and with%
\begin{equation*}
\bar{q}_{n,\tau ,j}^{\ast }(x;\eta _{1j})\equiv p_{n,\tau ,j}^{\ast
}(x)+\eta _{1j},
\end{equation*}%
where 
\begin{equation*}
p_{n,\tau ,j}^{\ast }(x)=\frac{1}{\sqrt{h^{d}}}\sum_{1\leq i\leq
N_{1}}\left\{ \beta _{n,x,\tau ,j}\left( Y_{ij}^{\ast },\frac{X_{i}^{\ast }-x%
}{h}\right) -\mathbf{E}\left[ \beta _{n,x,\tau ,j}\left( Y_{ij}^{\ast },%
\frac{X_{i}^{\ast }-x}{h}\right) \right] \right\} ,
\end{equation*}%
$\eta _{1j}$ is the $j$-th entry of $\eta _{1}$, and $N_{1}$ is a Poisson
random variable with mean $1$ and $((\eta _{1j})_{j\in A},N_{1})$ is
independent of $\{(Y_{i}^{\top },X_{i}^{\top },Y_{i}^{\ast \top
},X_{i}^{\ast \top })\}_{i=1}^{n}$. Let $[p_{n,\tau _{1}}^{\ast
(i)}(x),p_{n,\tau _{2}}^{\ast (i)}(x+uh)]$ be the i.i.d. copies of $%
[p_{n,\tau _{1}}^{\ast }(x),p_{n,\tau _{2}}^{\ast }(x+uh)]$ conditional on
the observations $\{(Y_{i},X_{i})\}_{i=1}^{n}$, and $\eta _{1}^{(i)}$ and $%
\eta _{2}^{(i)}$ be i.i.d. copies of $\eta _{1}$ and $\eta _{2}$. Define 
\begin{equation*}
q_{n,\tau _{1}}^{\ast (i)}(x;\eta _{1}^{(i)})=p_{n,\tau _{1}}^{\ast
(i)}(x)+\eta _{1}^{(i)}\text{ and }q_{n,\tau _{2}}^{\ast (i)}(x+uh;\eta
_{2}^{(i)})=p_{n,\tau _{2}}^{\ast (i)}(x+uh)+\eta _{2}^{(i)}.
\end{equation*}%
Note that%
\begin{equation*}
\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\left[ 
\begin{array}{c}
q_{n,\tau _{1}}^{\ast (i)}(x;\eta _{1}^{(i)}) \\ 
q_{n,\tau _{2}}^{\ast (i)}(x+uh;\eta _{2}^{(i)})%
\end{array}%
\right] =\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\left[ 
\begin{array}{c}
p_{n,\tau _{1}}^{\ast (i)}(x) \\ 
p_{n,\tau _{2}}^{\ast (i)}(x+uh)%
\end{array}%
\right] +\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\left[ 
\begin{array}{c}
\eta _{1}^{(i)} \\ 
\eta _{2}^{(i)}%
\end{array}%
\right] .
\end{equation*}%
The last sum has the same distribution as $[\eta _{1}^{\top },\eta
_{2}^{\top }]^{\top }$ and the leading sum on the right-hand side has the
same bootstrap distribution as that of $[\mathbf{z}_{N,\tau _{1}}^{\ast \top
}(x),\mathbf{z}_{N,\tau _{2}}^{\ast \top }(x+uh)]^{\top }$, $P$-a.e.
Therefore, we conclude that%
\begin{equation*}
\mathbf{\xi }_{N,\tau _{1},\tau _{2}}^{\ast }(x,u;\eta _{1}^{(i)},\eta
_{2}^{(i)})\overset{d^{\ast }}{=}\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\tilde{W}%
_{n,\tau _{1},\tau _{2}}^{(i)}(x,u;\eta _{1}^{(i)},\eta _{2}^{(i)}),
\end{equation*}%
where $\overset{d^{\ast }}{=}$ indicates the distributional equivalence with
respect to the product measure of the bootstrap distribution $P^{\ast }$ and
the joint distribution of $(\eta _{1}^{(i)},\eta _{2}^{(i)})$, $P$-a.e, and%
\begin{equation*}
\tilde{W}_{n,\tau _{1},\tau _{2}}^{(i)}(x,u;\eta _{1}^{(i)},\eta
_{2}^{(i)})\equiv \tilde{\Sigma}_{n,\tau _{1},\tau _{2},\bar{\varepsilon}%
}^{\ast -1/2}(x,u)\left[ 
\begin{array}{c}
q_{n}^{(i)}(x;\eta _{1}^{(i)}) \\ 
q_{n}^{(i)}(x+uh;\eta _{2}^{(i)})%
\end{array}%
\right] .
\end{equation*}%
Following the arguments in the proof of Lemma B6, we find that for each $%
u\in \mathcal{U},$ and for $\varepsilon \in (0,\varepsilon _{1})$ with $%
\varepsilon _{1}$ as in Assumption A6(i),%
\begin{eqnarray*}
&&\sup_{(x,u)\in (\mathcal{S}_{\tau _{1}}\cup \mathcal{S}_{\tau _{2}})\times 
\mathcal{U}}\sup_{P\in \mathcal{P}}\mathbf{E}\left[ \mathbf{E}^{\ast }||%
\tilde{W}_{n,\tau _{1},\tau _{2}}^{(i)}(x,u;\eta _{1}^{(i)},\eta
_{2}^{(i)})||^{3}\right] \\
&\leq &C_{1}\sup_{(x,u)\in (\mathcal{S}_{\tau _{1}}(\varepsilon )\cup 
\mathcal{S}_{\tau _{2}}(\varepsilon ))\times \mathcal{U}}\sup_{P\in \mathcal{%
P}}\mathbf{E}\left[ \lambda _{\min }^{3}\left( \tilde{\Sigma}_{n,\tau
_{1},\tau _{2},\bar{\varepsilon}}^{\ast -1/2}(x,u)\right) \mathbf{E}^{\ast
}||q_{n,\tau _{1}}^{\ast (i)}(x;\eta _{1}^{(i)})||^{3}\right] \\
&&+C_{1}\sup_{(x,u)\in (\mathcal{S}_{\tau _{1}}(\varepsilon )\cup \mathcal{S}%
_{\tau _{2}}(\varepsilon ))\times \mathcal{U}}\sup_{P\in \mathcal{P}}\mathbf{%
E}\left[ \lambda _{\min }^{3}\left( \tilde{\Sigma}_{n,\tau _{1},\tau _{2},%
\bar{\varepsilon}}^{\ast -1/2}(x,u)\right) \mathbf{E}^{\ast }||q_{n,\tau
_{2}}^{\ast (i)}(x+uh;\eta _{2}^{(i)})||^{3}\right] ,
\end{eqnarray*}%
for some $C_{1}>0$. As for the leading term,%
\begin{eqnarray*}
&&\sup_{(x,u)\in (\mathcal{S}_{\tau _{1}}(\varepsilon )\cup \mathcal{S}%
_{\tau _{2}}(\varepsilon ))\times \mathcal{U}}\sup_{P\in \mathcal{P}}\mathbf{%
E}\left[ \lambda _{\min }^{3}\left( \tilde{\Sigma}_{n,\tau _{1},\tau _{2},%
\bar{\varepsilon}}^{\ast -1/2}(x,u)\right) \mathbf{E}^{\ast }||q_{n,\tau
_{1}}^{\ast (i)}(x;\eta _{1}^{(i)})||^{3}\right] \\
&\leq &\sup_{(x,u)\in (\mathcal{S}_{\tau _{1}}(\varepsilon )\cup \mathcal{S}%
_{\tau _{2}}(\varepsilon ))\times \mathcal{U}}\sup_{P\in \mathcal{P}}\sqrt{%
\mathbf{E}\left[ \left( \mathbf{E}^{\ast }||q_{n,\tau _{1}}^{\ast
(i)}(x;\eta _{1}^{(i)})||^{3}\right) ^{2}\right] } \\
&&\times \sup_{(x,u)\in (\mathcal{S}_{\tau _{1}}(\varepsilon )\cup \mathcal{S%
}_{\tau _{2}}(\varepsilon ))\times \mathcal{U}}\sup_{P\in \mathcal{P}}\sqrt{%
\mathbf{E}\left[ \lambda _{\min }^{6}\left( \tilde{\Sigma}_{n,\tau _{1},\tau
_{2},\bar{\varepsilon}}^{\ast -1/2}(x,u)\right) \right] }\leq \frac{C_{2}%
\bar{\varepsilon}^{-3}}{\sqrt{h^{d}}},
\end{eqnarray*}%
by Lemma C5 and (\ref{lambda_min_b}). Similarly, we observe that 
\begin{equation*}
\sup_{(x,u)\in (\mathcal{S}_{\tau _{1}}(\varepsilon )\cup \mathcal{S}_{\tau
_{2}}(\varepsilon ))\times \mathcal{U}}\sup_{P\in \mathcal{P}}\mathbf{E}%
\left[ \lambda _{\min }^{3}\left( \tilde{\Sigma}_{n,\tau _{1},\tau _{2},\bar{%
\varepsilon}}^{\ast -1/2}(x,u)\right) \mathbf{E}^{\ast }||q_{n,\tau
_{2}}^{\ast (i)}(x+uh;\eta _{2}^{(i)})||^{3}\right] \leq \frac{C_{2}\bar{%
\varepsilon}^{-3}}{\sqrt{h^{d}}}.
\end{equation*}

Define 
\begin{equation*}
c_{n,\tau _{1},\tau _{2}}(x,u)=\tilde{C}_{n,p}\left( \frac{1}{\sqrt{n}}%
\sum_{i=1}^{n}\tilde{W}_{n,\tau _{1},\tau _{2}}^{(i)}(x,u;\eta
_{1}^{(i)},\eta _{2}^{(i)})\right) .
\end{equation*}%
Let $\Phi _{n,\tau _{1},\tau _{2}}(\cdot ;x,u)$ be the joint CDF of the
random vector $(\mathbb{\tilde{Z}}_{n,\tau _{1},\tau _{2},\bar{\varepsilon}%
}^{\top }(x),\mathbb{\tilde{Z}}_{n,\tau _{1},\tau _{2},\bar{\varepsilon}%
}^{\top }(x+uh))^{\top }$. By Theorem 1 of \citeasnoun{Sweeting:77},%
\begin{eqnarray}
&&\sup_{(x,u)\in (\mathcal{S}_{\tau _{1}}(\varepsilon )\cup \mathcal{S}%
_{\tau _{2}}(\varepsilon ))\times \mathcal{U}}\sup_{P\in \mathcal{P}}\mathbf{%
E}\left[ \left\vert c_{n,\tau _{1},\tau _{2}}(x,u)-\int \tilde{C}%
_{n,p}(\zeta )d\Phi _{n,\tau _{1},\tau _{2}}(\zeta ;x,u)\right\vert \right]
\label{bd0} \\
&\leq &\frac{C_{1}}{\sqrt{n}}\sup_{(x,u)\in (\mathcal{S}_{\tau
_{1}}(\varepsilon )\cup \mathcal{S}_{\tau _{2}}(\varepsilon ))\times 
\mathcal{U}}\sup_{P\in \mathcal{P}}\mathbf{E}\left[ \mathbf{E}^{\ast }||%
\tilde{W}_{n,\tau _{1},\tau _{2}}^{(i)}(x,u;\eta _{1}^{(i)},\eta
_{2}^{(i)})||^{3}\right] \leq \frac{C_{2}\bar{\varepsilon}^{-3}}{\sqrt{nh^{d}%
}}.  \notag
\end{eqnarray}%
Hence%
\begin{eqnarray*}
&&\mathbf{E}\left[ \left\vert \int_{B_{\tau _{1}}}\int_{\mathcal{U}}\left\{ 
\tilde{g}_{1n,\tau _{1},\tau _{2},\bar{\varepsilon}}(x,u)-\tilde{g}_{2n,\tau
_{1},\tau _{2},\bar{\varepsilon}}(x,u)\right\} w_{\tau _{1},B}(x)w_{\tau
_{2},B^{\prime }}(x+uh)dudx\right\vert \right] \\
&\leq &\int_{B_{\tau _{1}}}\int_{\mathcal{U}}\mathbf{E}\left\vert \tilde{g}%
_{1n,\tau _{1},\tau _{2},\bar{\varepsilon}}(x,u)-\tilde{g}_{2n,\tau
_{1},\tau _{2},\bar{\varepsilon}}(x,u)\right\vert w_{\tau _{1},B}(x)w_{\tau
_{2},B^{\prime }}(x+uh)dudx \\
&\leq &\int_{B_{\tau _{1}}}w_{\tau _{1},B}(x)w_{\tau _{2},B^{\prime }}(x)dx
\\
&&\times \sup_{(x,u)\in (\mathcal{S}_{\tau _{1}}(\varepsilon )\cup \mathcal{S%
}_{\tau _{2}}(\varepsilon ))\times \mathcal{U}}\sup_{P\in \mathcal{P}}%
\mathbf{E}\left\vert \tilde{g}_{1n,\tau _{1},\tau _{2},\bar{\varepsilon}%
}(x,u)-\tilde{g}_{2n,\tau _{1},\tau _{2},\bar{\varepsilon}}(x,u)\right\vert
\\
&\rightarrow &0,
\end{eqnarray*}%
as $n\rightarrow \infty $. The last convergence is due to (\ref{bd0}) and
hence uniform over $(\tau _{1},\tau _{2})\in \mathcal{T}\times \mathcal{T}$.
The proof of Step 1 is thus complete.

We turn to the second statement of Step 1. Similarly as in the proof of
Step 1 in the proof of Lemma B6, the second statement of Step 1 follows by
Lemma C4.

Now we turn to Step 2. In view of the proof of Step 2 in the proof of
Lemma B6, it suffices to show that with $s=(p+1)/(p-1)$ if $p>1$ and $s=2$
if $p=1,$%
\begin{eqnarray}
\sup_{\tau \in \mathcal{T},x\in \mathcal{S}_{\tau }(\varepsilon )}\sup_{P\in 
\mathcal{P}}\mathbf{E}\left[ \mathbf{E}^{\ast }\left\Vert \sqrt{nh^{d}}%
\mathbf{z}_{N,\tau }^{\ast }(x)\right\Vert ^{2s(p-1)}\right] &<&C\text{ and}
\label{bdd3} \\
\sup_{\tau \in \mathcal{T},x\in \mathcal{S}_{\tau }(\varepsilon )}\sup_{P\in 
\mathcal{P}}\mathbf{E}\left[ \mathbf{E}^{\ast }\left\Vert \sqrt{nh^{d}}%
\mathbf{z}_{N,\tau }^{\ast }(x;\eta _{1})\right\Vert ^{2s(p-1)}\right] &<&C%
\text{,}  \notag
\end{eqnarray}%
for some $C>0$. First note that for any $q>0$,%
\begin{eqnarray*}
&&\sup_{\tau \in \mathcal{T},x\in \mathcal{S}_{\tau }(\varepsilon
)}\sup_{P\in \mathcal{P}}\mathbf{E}\left[ \mathbf{E}^{\ast }\left\Vert \sqrt{%
nh^{d}}\{\mathbf{z}_{N,\tau }^{\ast }(x)-\mathbf{z}_{N,\tau }^{\ast }(x;\eta
_{1})\}\right\Vert ^{2q}\right] \\
&=&\mathbf{E}\left\Vert \sqrt{\bar{\varepsilon}}\mathbb{Z}\right\Vert ^{2q}=C%
\bar{\varepsilon}^{q}\text{,}
\end{eqnarray*}%
where $\mathbb{Z}\in \mathbf{R}^{J}$ is a centered normal random vector with
covariance matrix $I_{J}$. Also, we deduce that for some constants $%
C_{1},C_{2}>0$,%
\begin{eqnarray*}
&&\sup_{\tau \in \mathcal{T},x\in \mathcal{S}_{\tau }(\varepsilon
)}\sup_{P\in \mathcal{P}}\mathbf{E}\left[ \mathbf{E}^{\ast }\left\Vert \sqrt{%
nh^{d}}\mathbf{z}_{N,\tau }^{\ast }(x)\right\Vert ^{2s(p-1)}\right] \\
&\leq &\sup_{\tau \in \mathcal{T},x\in \mathcal{S}_{\tau }(\varepsilon
)}\sup_{P\in \mathcal{P}}\mathbf{E}\left[ \mathbf{E}^{\ast }\left\Vert \sqrt{%
nh^{d}}\mathbf{z}_{N,\tau }^{\ast }(x;\eta _{1})\right\Vert ^{2s(p-1)}\right]
+C_{1}\bar{\varepsilon}^{s(p-1)}\leq C_{1}+C_{2}\bar{\varepsilon}^{s(p-1)},
\end{eqnarray*}%
by the third statement of Lemma C5. This leads to the first and second
statements of (\ref{bdd3}). Thus the proof of the lemma is complete.
\end{proof}

\begin{LemmaD}
\label{lem-d7} \textit{Suppose that for some small }$\nu _{1}>0$, $%
n^{-1/2}h^{-d-\nu _{1}}\rightarrow 0 $\textit{, as }$n\rightarrow
\infty $\textit{\ and the conditions of Lemma B6 hold. Then there exists} $%
C>0$\textit{\ such that for any sequence of Borel sets }$B_{n}\subset 
\mathcal{S},$ \textit{and }$A\subset \mathbb{N}_{J},$\textit{\ from some
large }$n$\textit{\ on,}%
\begin{align*}
& \sup_{P\in \mathcal{P}}\mathbf{E}\left( \mathbf{E}^{\ast }\left[
\left\vert h^{-d/2}\int_{B_{n}}\left\{ \Lambda _{A,p}(\sqrt{nh^{d}}\mathbf{z}%
_{n,\tau }^{\ast }(x))-\mathbf{E}^{\ast }\left[ \Lambda _{A,p}(\sqrt{nh^{d}}%
\mathbf{z}_{N,\tau }^{\ast }(x))\right] \right\} dQ(x,\tau )\right\vert %
\right] \right) \\
& \leq C\sqrt{Q(B_{n})}.
\end{align*}
\end{LemmaD}

\begin{proof}[Proof of Lemma C\protect\ref{lem-d7}]
We follow the proof of Lemma B7 and show that for some $C>0$,

\textbf{Step 1:} $\sup_{P\in \mathcal{P}}\mathbf{E}\left( \mathbf{E}^{\ast }%
\left[ \left\vert h^{-d/2}\int_{B_{n}}\left\{ \Lambda _{A,p}(\sqrt{nh^{d}}%
\mathbf{z}_{n,\tau }^{\ast }(x))-\Lambda _{A,p}(\sqrt{nh^{d}}\mathbf{z}%
_{N,\tau }^{\ast }(x))\right\} dQ(x,\tau )\right\vert \right] \right) \leq
CQ(B_{n}),$ and

\textbf{Step 2:}%
\begin{eqnarray*}
&&\sup_{P\in \mathcal{P}}\mathbf{E}\left( \mathbf{E}^{\ast }\left[
\left\vert h^{-d/2}\int_{B_{n}}\left\{ \Lambda _{A,p}(\sqrt{nh^{d}}\mathbf{z}%
_{N,\tau }^{\ast }(x))-\mathbf{E}^*[\Lambda _{A,p}(\sqrt{nh^{d}}\mathbf{z}_{N,\tau
}^{\ast }(x))]\right\} dQ(x,\tau )\right\vert \right] \right) \\
&\leq &C\sqrt{Q(B_{n})}.
\end{eqnarray*}

\noindent \textbf{Proof of Step 1: }Similarly as in the proof of Step 1 in
the proof of Lemma B7, we first write%
\begin{equation*}
\mathbf{z}_{n,\tau }^{\ast }(x)=\mathbf{z}_{N,\tau }^{\ast }(x)+\mathbf{v}%
_{n,\tau }^{\ast }(x)+\mathbf{s}_{n,\tau }^{\ast }(x),
\end{equation*}%
where%
\begin{eqnarray*}
\mathbf{v}_{n,\tau }^{\ast }(x) &\equiv &\left( \frac{n-N}{n}\right) \cdot 
\frac{1}{h^{d}}\mathbf{E}^{\ast }\left[ \beta _{n,x,\tau }\left( Y_{i}^{\ast
},\frac{X_{i}^{\ast }-x}{h}\right) \right] \text{ and} \\
\mathbf{s}_{n,\tau }^{\ast }(x) &\equiv &\frac{1}{nh^{d}}\sum_{i=N+1}^{n}%
\left\{ \beta _{n,x,\tau }\left( Y_{i}^{\ast },\frac{X_{i}^{\ast }-x}{h}%
\right) -\mathbf{E}^{\ast }\left[ \beta _{n,x,\tau }\left( Y_{i}^{\ast },%
\frac{X_{i}^{\ast }-x}{h}\right) \right] \right\} .
\end{eqnarray*}

Similarly as in the proof of Lemma B7, we deduce that for some $%
C_{1},C_{2}>0,$%
\begin{eqnarray*}
\lefteqn{\left\vert \int_{B_{n}}\left\{ \Lambda _{A,p}\left( \mathbf{z}%
_{n,\tau }^{\ast }(x)\right) -\Lambda _{A,p}\left( \mathbf{z}_{N,\tau
}^{\ast }(x)\right) \right\} dQ(x,\tau )\right\vert } \\
&\leq &C_{1}\int_{B_{n}}\left\Vert \mathbf{v}_{n,\tau }^{\ast
}(x)\right\Vert \left( \left\Vert \mathbf{z}_{n,\tau }^{\ast }(x)\right\Vert
^{p-1}+\left\Vert \mathbf{z}_{N,\tau }^{\ast }(x)\right\Vert ^{p-1}\right)
dQ(x,\tau ) \\
&&+C_{2}\int_{B_{n}}\left\Vert \mathbf{s}_{n,\tau }^{\ast }(x)\right\Vert
\left( \left\Vert \mathbf{z}_{n,\tau }^{\ast }(x)\right\Vert
^{p-1}+\left\Vert \mathbf{z}_{N,\tau }^{\ast }(x)\right\Vert ^{p-1}\right)
dQ(x,\tau ) \\
&=&D_{1n}^{\ast }+D_{2n}^{\ast },\text{ say.}
\end{eqnarray*}%
To deal with $D_{1n}^{\ast }$ and $D_{2n}^{\ast }$, we first show the
following:\medskip

\noindent \textsc{Claim 1:} $\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in 
\mathcal{P}}\mathbf{E}\left( \mathbf{E}^{\ast }[||\mathbf{v}_{n,\tau }^{\ast
}(x)||^{2}]\right) =O(n^{-1}).$\medskip

\noindent \textsc{Claim 2:} $\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in 
\mathcal{P}}\mathbf{E}\left( \mathbf{E}^{\ast }[||\mathbf{s}_{n,\tau }^{\ast
}(x)||^{2}]\right) =O(n^{-3/2}h^{-d})$.\medskip

\noindent \textsc{Proof of Claim 1:} Similarly as in the proof of Lemma B7,
we note that%
\begin{equation*}
\mathbf{E}\left( \mathbf{E}^{\ast }[||\mathbf{v}_{n,\tau }^{\ast
}(x)||^{2}]\right) \leq \mathbf{E}\left\vert \left( \frac{n-N}{n}\right)
\right\vert ^{2}\mathbf{E}\left[ \left\Vert \frac{1}{h^{d}}\mathbf{E}^{\ast }%
\left[ \beta _{n,x,\tau }\left( Y_{i}^{\ast },\frac{X_{i}^{\ast }-x}{h}%
\right) \right] \right\Vert ^{2}\right] .
\end{equation*}%
By the first statement of Lemma C5, we have%
\begin{equation*}
\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\mathbf{E}\left[
\left\Vert \frac{1}{h^{d}}\mathbf{E}^{\ast }\left[ \beta _{n,x,\tau }\left(
Y_{i}^{\ast },\frac{X_{i}^{\ast }-x}{h}\right) \right] \right\Vert ^{2}%
\right] =O(1).
\end{equation*}%
Since $\mathbf{E}\left\vert (n-N)/n\right\vert ^{2}=O(n^{-1})$, we obtain
Claim 1.\medskip

\noindent \textsc{Proof of Claim 2:} Let 
\begin{equation*}
\mathbf{s}_{n,\tau }^{\ast }(x;\eta _{1})=\mathbf{s}_{n,\tau }^{\ast }(x)+%
\frac{(N-n)\eta _{1}}{n^{3/2}h^{d/2}},
\end{equation*}%
where $\eta _{1}$ is a random vector independent of $((Y_{i}^{\ast
},X_{i}^{\ast })_{i=1}^{n},(Y_{i},X_{i})_{i=1}^{n},N)$ and follows $N(0,\bar{%
\varepsilon}I_{J})$. Note that%
\begin{eqnarray*}
&&\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\mathbf{E}\left( 
\mathbf{E}^{\ast }\left\Vert \sqrt{nh^{d}}\mathbf{s}_{n,\tau }^{\ast
}(x)\right\Vert ^{2}\right) \\
&\leq &2\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\mathbf{E}%
\left( \mathbf{E}^{\ast }\left\Vert \sqrt{nh^{d}}\mathbf{s}_{n,\tau }^{\ast
}(x;\eta _{1})\right\Vert ^{2}\right) +\frac{2}{n}\mathbf{E}\left\Vert \frac{%
(N-n)\eta _{1}}{\sqrt{n}}\right\Vert ^{2} \\
&\leq &2\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\mathbf{E}%
\left( \mathbf{E}^{\ast }\left\Vert \sqrt{nh^{d}}\mathbf{s}_{n,\tau }^{\ast
}(x;\eta _{1})\right\Vert ^{2}\right) +\frac{C\bar{\varepsilon}^{2}}{n},
\end{eqnarray*}%
as in the proof of Lemma B7. As for the leading expectation on the right
hand side of (\ref{dec22}), we let $C_{1}>0$ be as in Lemma C4 and note that%
\begin{eqnarray*}
\mathbf{E}\left( \mathbf{E}^{\ast }\left\Vert \sqrt{nh^{d}}\mathbf{s}%
_{n,\tau }^{\ast }(x;\eta _{1})\right\Vert ^{2}\right) &=&\sum_{j\in \mathbb{%
N}_{J}}\mathbf{E}\left( \mathbf{E}^{\ast }\left( \frac{1}{\sqrt{n}}%
\sum_{i=N+1}^{n}q_{n,\tau ,j}^{\ast (i)}(x;\eta _{1j}^{(i)})\right)
^{2}\right) \\
&=&\frac{1}{n}\sum_{j\in \mathbb{N}_{J}}\mathbf{E}\left( \tilde{\sigma}%
_{n,\tau ,j}^{2}(x)\mathbf{E}^{\ast }\left( \sum_{i=N+1}^{n}\frac{q_{n,\tau
,j}^{\ast (i)}(x;\eta _{1j}^{(i)})}{\tilde{\sigma}_{n,\tau ,j}(x)}\right)
^{2}\right) ,
\end{eqnarray*}%
where $q_{n,\tau }^{\ast (i)}(x;\eta _{1}^{(i)})$'s ($i=1,2,...$) are as defined in the proof of Lemma C6 and $q_{n,\tau ,j}^{\ast
(i)}(x;\eta _{1j}^{(i)})$ is the $j$-th entry of $q_{n,\tau }^{\ast
(i)}(x;\eta _{1}^{(i)})$ and $\tilde{\sigma}_{n,\tau ,j}^{2}(x)=Var^{\ast
}(q_{n,\tau ,j}^{\ast (i)}(x;\eta _{1j}^{(i)}))>0$ and $Var^{\ast }$ denotes
the variance with respect to the joint distribution of $((Y_{i}^{\ast
},X_{i}^{\ast })_{i=1}^{n},\eta _{1j}^{(i)})$ conditional on $%
(Y_{i},X_{i})_{i=1}^{n}.$ We apply Lemma 1(i) of \citeasnoun{Horvath:91} to
deduce that%
\begin{eqnarray}
\mathbf{E}^{\ast }\left( \sum_{i=N+1}^{n}\frac{q_{n,\tau ,j}^{\ast
(i)}(x;\eta _{1j}^{(i)})}{\tilde{\sigma}_{n,\tau ,j}(x)}\right) ^{2} &\leq &C%
\sqrt{n}+C\mathbf{E}^{\ast }\left( \left\vert \frac{q_{n,\tau ,j}^{\ast
(i)}(x;\eta _{1j}^{(i)})}{\tilde{\sigma}_{n,\tau ,j}(x)}\right\vert
^{3}\right)  \label{dev4} \\
&&+C\mathbf{E}^{\ast }\left( \left\vert \frac{q_{n,\tau ,j}^{\ast
(i)}(x;\eta _{1j}^{(i)})}{\tilde{\sigma}_{n,\tau ,j}(x)}\right\vert
^{4}\right) ,  \notag
\end{eqnarray}%
for some $C>0$. Using this, Lemma C5, and following arguments similarly as
in (\ref{arg1}), (\ref{arg2}) and (\ref{arg3}), we conclude that 
\begin{eqnarray*}
\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\mathbf{E}\left( 
\mathbf{E}^{\ast }\left\Vert \sqrt{nh^{d}}\mathbf{s}_{n,\tau }^{\ast
}(x)\right\Vert ^{2}\right) &\leq &O\left( n^{-1}h^{-\nu _{1}}\right)
+O\left( n^{-1/2}+n^{-3/4}h^{-d/2-\nu _{1}}+n^{-1}h^{-d-\nu _{1}}\right) \\
&=&O\left( n^{-1}h^{-\nu _{1}}\right) +O\left( n^{-1/2}\right) ,
\end{eqnarray*}%
since $n^{-1/2}h^{-d-\nu _{1}}\rightarrow 0$. This delivers Claim 2.

Using Claims 1 and 2, and following the arguments in the proof of Lemma B7,
we obtain Step 1.\medskip

\noindent \textbf{Proof of Step 2:} We can follow the proof of Lemma B6 to
show that%
\begin{eqnarray*}
&&\mathbf{E}\left[ \mathbf{E}^{\ast }\left[ h^{-d/2}\int_{B_{n}}\left(
\Lambda _{A,p}(\sqrt{nh^{d}}\mathbf{z}_{N,\tau }^{\ast }(x))-\mathbf{E}%
^{\ast }\left[ \Lambda _{A,p}(\sqrt{nh^{d}}\mathbf{z}_{N,\tau }^{\ast }(x))%
\right] \right) dQ(x,\tau )\right] ^{2}\right] \\
&=&\mathbf{E}\left[ \int_{\mathcal{T}}\int_{\mathcal{T}}\int_{B_{n,\tau
_{1}}\cap B_{n,\tau _{2}}}\int_{\mathcal{U}}C_{n,\tau _{1},\tau
_{2},A,A^{\prime }}^{\ast }(x,u)dudxd\tau _{1}d\tau _{2}\right] +o(1) \\
&\leq &C\int_{\mathcal{T}}\int_{\mathcal{T}}\int_{B_{n,\tau _{1}}\cap
B_{n,\tau _{2}}}dxd\tau _{1}d\tau _{2}+o(1)\leq CQ(B_{n}),
\end{eqnarray*}%
where $C_{n,\tau _{1},\tau _{2},A,A^{\prime }}^{\ast }(x,v)$ is as defined
in (\ref{C*}). We obtain the desired result of Step 2.
\end{proof}

Let $\mathcal{C}\subset \mathbf{R}^{d},$ $\alpha _{P}\equiv P\{X\in \mathbf{R%
}^{d}\backslash \mathcal{C}\}$ and $B_{n,A}(c_{n};\mathcal{C})\ $be as
introduced prior to Lemma B8. Define%
\begin{eqnarray*}
\zeta _{n,A}^{\ast } &\equiv &\int_{B_{n,A}(c_{n};\mathcal{C})}\Lambda
_{A,p}(\sqrt{nh^{d}}\mathbf{z}_{n,\tau }^{\ast }(x))dQ(x,\tau ),\text{ and}
\\
\zeta _{N,A}^{\ast } &\equiv &\int_{B_{n,A}(c_{n};\mathcal{C})}\Lambda
_{A,p}(\sqrt{nh^{d}}\mathbf{z}_{N,\tau }^{\ast }(x))dQ(x,\tau ).
\end{eqnarray*}%
Let $\mu _{A}$'s be real numbers indexed by $A\subset \mathbb{N}_{J}$. We
also define $B_{n,A}(c_{n};\mathcal{C})$ as prior to Lemma B8 and let 
\begin{eqnarray*}
S_{n}^{\ast } &\equiv &h^{-d/2}\sum_{A\in \mathcal{N}_{J}}\mu _{A}\left\{
\zeta _{N,A}^{\ast }-\mathbf{E}^{\ast }\zeta _{N,A}^{\ast }\right\} , \\
U_{n}^{\ast } &\equiv &\frac{1}{\sqrt{n}}\left\{
\sum_{i=1}^{N}1\{X_{i}^{\ast }\in \mathcal{C}\}-nP^{\ast }\left\{
X_{i}^{\ast }\in \mathcal{C}\right\} \right\} ,\text{ and} \\
V_{n}^{\ast } &\equiv &\frac{1}{\sqrt{n}}\left\{
\sum_{i=1}^{N}1\{X_{i}^{\ast }\in \mathbf{R}^{d}\backslash \mathcal{C}%
\}-nP^{\ast }\left\{ X_{i}^{\ast }\in \mathbf{R}^{d}\backslash \mathcal{C}%
\right\} \right\} .
\end{eqnarray*}%
We let 
\begin{equation*}
H_{n}^{\ast }\equiv \left[ \frac{S_{n}^{\ast }}{\sigma _{n}(\mathcal{C})},%
\frac{U_{n}^{\ast }}{\sqrt{1-\alpha _{P}}}\right] .
\end{equation*}%
The following lemma is a bootstrap counterpart of Lemma B8.

\begin{LemmaD}
\label{lem-d8} \textit{Suppose that the conditions of Lemma C6 hold and that 
}$c_{n}\rightarrow \infty $, \textit{as }$n\rightarrow \infty . $

\noindent (i)\textit{\ If } $\liminf_{n\rightarrow \infty }\inf_{P\in 
\mathcal{P}}\sigma _{n}^{2}(\mathcal{C})>0$, \textit{then for all }$a>0,$ 
\begin{equation*}
\sup_{P\in \mathcal{P}}P\left\{ \sup_{t\in \mathbf{R}^{2}}\left\vert P^{\ast
}\left\{ H_{n}^{\ast }\leq t\right\} -P\left\{ \mathbb{Z}\leq t\right\}
\right\vert >a\right\} \rightarrow 0,
\end{equation*}
\textit{where }$\mathbb{Z} \sim N(0,I_2)$.

\noindent (ii) \textit{If } $\limsup_{n\rightarrow \infty }\sigma _{n}^{2}(%
\mathcal{C})=0$, \textit{then, for each }$(t_{1},t_{2})\in \mathbf{R}^{2}$ 
\textit{and }$a>0,$ 
\begin{equation*}
\sup_{P\in \mathcal{P}}P\left\{ \left\vert P^{\ast }\left\{ S_{n}^{\ast
}\leq t_{1}\text{ and }\frac{U_{n}^{\ast }}{\sqrt{1-\alpha _{P}}}\leq
t_{2}\right\} -1\left\{ 0\leq t_{1}\right\} P\left\{ \mathbb{Z}_{1}\leq
t_{2}\right\} \right\vert >a\right\} \rightarrow 0.
\end{equation*}
\end{LemmaD}

\begin{proof}[Proof of Lemma C\protect\ref{lem-d8}]
Similarly as in the proof of Lemma C8, we fix $\bar{\varepsilon}>0$ and let%
\begin{equation*}
H_{n,\bar{\varepsilon}}^{\ast }\equiv \left[ \frac{S_{n,\bar{\varepsilon}%
}^{\ast }}{\sigma _{n,\bar{\varepsilon}}(\mathcal{C})},\frac{U_{n}^{\ast }}{%
\sqrt{1-\alpha _{P}}}\right] ^{\top },
\end{equation*}%
where $S_{n,\bar{\varepsilon}}^{\ast }$ is equal to $S_{n}^{\ast }$, except
that $\zeta _{N,A}^{\ast }$ is replaced by%
\begin{equation*}
\zeta _{N,A,\bar{\varepsilon}}^{\ast }\equiv \int_{B_{n,A}(c_{n};\mathcal{C}%
)}\Lambda _{A,p}(\sqrt{nh^{d}}\mathbf{z}_{N,\tau }^{\ast }(x;\eta
_{1}))dQ(x,\tau ),
\end{equation*}%
and $\mathbf{z}_{N,\tau }^{\ast }(x;\eta _{1})$ is as defined prior to Lemma
C6. Also let 
\begin{equation*}
\tilde{C}_{n}\equiv \mathbf{E}^{\ast }H_{n}^{\ast }H_{n}^{\ast \top }\text{
and }\tilde{C}_{n,\bar{\varepsilon}}\equiv \mathbf{E}^{\ast }H_{n,\bar{%
\varepsilon}}^{\ast }H_{n,\bar{\varepsilon}}^{\ast \top }\text{.}
\end{equation*}%
First, we show the following statements.\medskip

\noindent \textbf{Step 1: }sup$_{P\in \mathcal{P}}P\left\{ |Cov^{\ast }(S_{n,%
\bar{\varepsilon}}^{\ast }-S_{n}^{\ast },U_{n}^{\ast })|>M\sqrt{\bar{%
\varepsilon}}\right\} \rightarrow 0,$ as $n\rightarrow \infty $ and $%
M\rightarrow \infty $.\medskip

\noindent \textbf{Step 2: }For any $a>0$, sup$_{P\in \mathcal{P}}P\left\{
\left\vert Cov(S_{n,\bar{\varepsilon}}^{\ast },U_{n}^{\ast })\right\vert
>ah^{d/2}\right\} \rightarrow 0,$ as $n\rightarrow \infty .$\medskip

\noindent \textbf{Step 3:\ }There exists $c>0$ such that from some large $n$
on,%
\begin{equation*}
\inf_{P\in \mathcal{P}}\lambda _{\min }(\tilde{C}_{n})>c.
\end{equation*}

\noindent \textbf{Step 4: }For any $a>0$, as $n\rightarrow \infty ,$%
\begin{equation*}
\sup_{P\in \mathcal{P}}P\left\{ \sup_{t\in \mathbf{R}^{2}}\left\vert P^{\ast
}\left\{ \tilde{C}_{n}^{-1/2}H_{n}^{\ast }\leq t\right\} \rightarrow
P\left\{ \mathbb{Z}\leq t\right\} \right\vert >a\right\} \rightarrow 0\text{.%
}
\end{equation*}

Combining Steps 1-4, we obtain (i) of Lemma B8.\medskip

\noindent \textbf{Proof of Step 1:} Observe that 
\begin{equation*}
\left\vert \zeta _{N,A,\bar{\varepsilon}}^{\ast }-\zeta _{N,A}^{\ast
}\right\vert \leq C||\eta _{1}||\int_{B_{n,A}(c_{n};\mathcal{C})}\left\Vert 
\sqrt{nh^{d}}\mathbf{z}_{N,\tau }^{\ast }(x)\right\Vert ^{p-1}dQ(x,\tau ).
\end{equation*}%
As in the proof of Step 1 in the proof of Lemma B8, we deduce that 
\begin{equation*}
\mathbf{E}^{\ast }\left[ \left\vert \zeta _{N,A,\bar{\varepsilon}}^{\ast
}-\zeta _{N,A}^{\ast }\right\vert ^{2}\right] \leq C\bar{\varepsilon}%
\int_{B_{n,A}(c_{n};\mathcal{C})}\mathbf{E}^{\ast }\left\Vert \sqrt{nh^{d}}%
\mathbf{z}_{N,\tau }^{\ast }(x)\right\Vert ^{2p-2}dQ(x,\tau ).
\end{equation*}%
Hence for some $C_{1},C_{2}>0,$%
\begin{eqnarray}
&&\mathbf{E}\left( \mathbf{E}^{\ast }\left[ \left\vert \zeta _{N,A,\bar{%
\varepsilon}}^{\ast }-\zeta _{N,A}^{\ast }\right\vert ^{2}\right] \right)
\label{dev6} \\
&\leq &C\bar{\varepsilon}\int_{B_{n,A}(c_{n};\mathcal{C})}\mathbf{E}\left( 
\mathbf{E}^{\ast }\left\Vert \sqrt{nh^{d}}\mathbf{z}_{N,\tau }^{\ast
}(x)\right\Vert ^{2p-2}\right) dQ(x,\tau )\leq C_{2}\bar{\varepsilon}  \notag
\end{eqnarray}%
by the second statement of Lemma C5.

On the other hand, observe that $\mathbf{E}^{\ast }U_{n}^{\ast 2}\leq 1.$
Hence%
\begin{equation*}
P\left\{ |Cov^{\ast }(S_{n,\bar{\varepsilon}}^{\ast }-S_{n}^{\ast
},U_{n}^{\ast })|>M\sqrt{\bar{\varepsilon}}\right\} \leq |\mathcal{N}%
_{J}|\cdot P\left\{ \max_{A\in \mathcal{N}_{J}}\mathbf{E}^{\ast }\left[
\left\vert \zeta _{N,A,\bar{\varepsilon}}^{\ast }-\zeta _{N,A}^{\ast
}\right\vert ^{2}\right] >M^{2}\bar{\varepsilon}\right\} .
\end{equation*}%
By Markov's inequality, the last probability is bounded by (for some $C>0$
that does not depend on $P\in \mathcal{P}$)%
\begin{equation*}
M^{-2}\bar{\varepsilon}^{-1}\sum_{A\in \mathcal{N}_{J}}\mathbf{E}\left( 
\mathbf{E}^{\ast }\left[ \left\vert \zeta _{N,A,\bar{\varepsilon}}^{\ast
}-\zeta _{N,A}^{\ast }\right\vert ^{2}\right] \right) \leq CM^{-2},
\end{equation*}%
by (\ref{dev6}). Hence we obtain the desired result.\medskip

\noindent \textbf{Proof of Step 2:} Let $\tilde{\Sigma}_{2n,\tau ,\bar{%
\varepsilon}}^{\ast }$ be the covariance matrix of $[(q_{n,\tau }^{\ast
}(x)+\eta _{1})^{\top },\tilde{U}_{n}^{\ast }]^{\top }$ under $P^{\ast }$,
where $\tilde{U}_{n}^{\ast }=U_{n}^{\ast }/\sqrt{P\{X\in \mathcal{C}\}}.$
Using Lemma C4 and following the same arguments in (\ref{dev7}), we find that%
\begin{equation*}
\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\mathbf{E}\left[ 
\mathbf{E}^{\ast }\left[ q_{n,\tau ,j}^{\ast }(x)\tilde{U}_{n}^{\ast }\right]
\right] \leq C_{2}h^{d/2},
\end{equation*}%
for some $C_{2}>0$. Therefore, using this result and following the proof of
Step 3 in the proof of Lemma B8, we deduce that (everywhere)%
\begin{equation}
\lambda _{\min }\left( \tilde{\Sigma}_{2n,\tau ,\bar{\varepsilon}}^{\ast
}\right) \geq \bar{\varepsilon}-\left\Vert A_{n,\tau }^{\ast }(x)\right\Vert
,  \label{ineq37}
\end{equation}%
for some random matrix $A_{n,\tau }^{\ast }(x)$ such that 
\begin{equation*}
\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\mathbf{E}\left[
\left\Vert A_{n,\tau }^{\ast }(x)\right\Vert \right] =O(h^{d/2}).
\end{equation*}%
Hence by (\ref{ineq37}), 
\begin{eqnarray}
&&\inf_{(x,\tau )\in \mathcal{S}}\inf_{P\in \mathcal{P}}P\left\{ \lambda
_{\min }\left( \tilde{\Sigma}_{2n,\tau ,\bar{\varepsilon}}^{\ast }\right)
\geq \bar{\varepsilon}/2\right\}  \label{lb3} \\
&\geq &\inf_{(x,\tau )\in \mathcal{S}}\inf_{P\in \mathcal{P}}P\left\{
\left\Vert A_{n,\tau }^{\ast }(x)\right\Vert \leq \bar{\varepsilon}/2\right\}
\notag \\
&\geq &1-\frac{2}{\bar{\varepsilon}}\sup_{(x,\tau )\in \mathcal{S}%
}\sup_{P\in \mathcal{P}}\mathbf{E}\left[ \left\Vert A_{n,\tau }^{\ast
}(x)\right\Vert \right] \rightarrow 1,  \notag
\end{eqnarray}%
as $n\rightarrow \infty $.

Now note that 
\begin{equation*}
\left( q_{n,\tau ,j}^{\ast }(x),\tilde{U}_{n}^{\ast }\right) \overset{%
d^{\ast }}{=}\left( \frac{1}{\sqrt{n}}\sum_{k=1}^{n}q_{n,\tau ,j}^{(k)\ast
}(x),\frac{1}{\sqrt{n}}\sum_{k=1}^{n}\tilde{U}_{n}^{(k)\ast }\right) ,
\end{equation*}%
where $(q_{n,\tau ,j}^{(k)\ast }(x),\tilde{U}_{n}^{(k)\ast })$'s with $%
k=1,...,n$ are i.i.d. copies of $(q_{n,\tau ,j}^{\ast }(x),%
\bar{U}_{n}^{\ast })$, and 
\begin{equation*}
\bar{U}_{n}^{\ast }\equiv \frac{1}{\sqrt{nP\{X\in \mathcal{C}\}}}\left\{
\sum_{1\leq i\leq N_{1}}1\{X_{i}^{\ast }\in \mathcal{C}\}-P^{\ast }\left\{
X_{i}^{\ast }\in \mathcal{C}\right\} \right\} .
\end{equation*}%
Note also that by Rosenthal's inequality, 
\begin{equation*}
\text{limsup}_{n\rightarrow \infty }\sup_{P\in \mathcal{P}}P\left\{ \mathbf{E%
}^{\ast }\left[ |\tilde{U}_{n}^{(k)\ast }|^{3}\right] >M\right\} \rightarrow
0,
\end{equation*}%
as $M\rightarrow \infty $. Define%
\begin{equation*}
W_{n,\tau }^{\ast }(x;\eta _{1})\equiv \tilde{\Sigma}_{2n,\tau ,\bar{%
\varepsilon}}^{\ast -1/2}\left[ 
\begin{array}{c}
q_{n,\tau }^{\ast }(x)+\eta _{1} \\ 
\tilde{U}_{n}^{\ast }%
\end{array}%
\right] .
\end{equation*}%
Using (\ref{lb3}) and Lemma C5, and following the same arguments in the
proof of Step 2 in the proof of Lemma B8, we deduce that%
\begin{equation*}
\text{limsup}_{n\rightarrow \infty }\sup_{(x,\tau )\in \mathcal{S}%
}\sup_{P\in \mathcal{P}}P\left\{ \mathbf{E}^{\ast }\left\Vert W_{n,\tau
}^{\ast }(x;\eta _{1})\right\Vert ^{3}>M\bar{\varepsilon}^{-3/2}h^{-d/2}%
\right\} \rightarrow 0,
\end{equation*}%
as $M\rightarrow \infty $. For any vector $\mathbf{v}=[\mathbf{v}_{1}^{\top
},v_{2}]^{\top }\in \mathbf{R}^{J+1}$, we define%
\begin{equation*}
\tilde{D}_{n,\tau ,p}(\mathbf{v})\equiv \Lambda _{p}\left( \left[ \tilde{%
\Sigma}_{2n,\tau ,\bar{\varepsilon}}^{\ast 1/2}\mathbf{v}\right] _{1}\right) %
\left[ \tilde{\Sigma}_{2n,\tau ,\bar{\varepsilon}}^{\ast 1/2}\mathbf{v}%
\right] _{2},
\end{equation*}%
where $[a]_{1}$ of a vector $a\in \mathbf{R}^{J+1}$ indicates the vector of
the first $J$ entries of $a$, and $[a]_{2}$ the last entry of $a.$ By
Theorem 1 of \citeasnoun{Sweeting:77}, we find that (with $\bar{\varepsilon}%
>0$ fixed)%
\begin{equation*}
\mathbf{E}^{\ast }\left[ \tilde{D}_{n,\tau ,p}\left( \frac{1}{\sqrt{n}}%
\sum_{i=1}^{n}W_{n,\tau }^{(i)\ast }(x;\eta _{1})\right) \right] =\mathbf{E}%
\left[ \tilde{D}_{n,\tau ,p}\left( \mathbb{Z}_{J+1}\right) \right]
+O_{P}(n^{-1/2}h^{-d/2}) = o_P(n^{d/2}),
\end{equation*}%
$\mathcal{P}$-uniformly, where $\mathbb{Z}_{J+1}\sim N(0,I_{J+1})$ and $W_{n,\tau }^{(i)\ast }(x;\eta
_{1})$'s are i.i.d. copies of $W_{n,\tau }^{\ast }(x;\eta _{1})$ under $%
P^{\ast }$. The last equality follows because $n^{-1/2}h^{-d/2}=o(h^{d/2})$ and $\mathbf{E}[\tilde{D}%
_{n,\tau ,p}\left( \mathbb{Z}_{J+1}\right) ]=0$. Since
\begin{equation*}
Cov^{\ast }\left( \Lambda _{A,p}\left( \sqrt{nh^{d}}\mathbf{z}_{N,\tau
}^{\ast }(x;\eta _{1})\right) ,U_{n}^{\ast }\right) =\mathbf{E}^{\ast }\left[
\tilde{D}_{n,\tau ,p}\left( \frac{1}{\sqrt{n}}\sum_{i=1}^{n}W_{n,\tau
}^{(i)\ast }(x)\right) \right],
\end{equation*}%
we conclude that 
\begin{equation}
\sup_{(x,\tau )\in \mathcal{S}}\left\vert Cov^{\ast }\left( \Lambda
_{A,p}\left( \sqrt{nh^{d}}\mathbf{z}_{N,\tau }^{\ast }(x;\eta _{1})\right)
,U_{n}^{\ast }\right) \right\vert =o_{P}(h^{d/2}),  \label{cv11}
\end{equation}%
uniformly in $P\in \mathcal{P}$.

Now for some $C>0,$%
\begin{equation*}
P\left\{ \left\vert Cov(S_{n,\bar{\varepsilon}}^{\ast },U_{n}^{\ast
})\right\vert >ah^{d/2}\right\} \leq P\left\{ C\sup_{(x,\tau )\in \mathcal{S}%
}\left\vert Cov^{\ast }\left( \Lambda _{A,p}\left( \sqrt{nh^{d}}\mathbf{z}%
_{N,\tau }^{\ast }(x;\eta _{1})\right) ,U_{n}^{\ast }\right) \right\vert
>ah^{d/2}\right\} .
\end{equation*}%
The last probability vanishes uniformly in $P\in \mathcal{P}$ by (\ref{cv11}%
). By applying the Dominated Convergence Theorem, we obtain Step 2.\medskip

\noindent \textbf{Proof of Step 3:} First, we show that 
\begin{equation}
Var^{\ast }\left( S_{n}^{\ast }\right) =\sigma _{n}^{2}(\mathcal{C}%
)+o_{P}(1),  \label{varvar}
\end{equation}%
where $o_{P}(1)$ is uniform over $P\in \mathcal{P}$. Note that%
\begin{equation*}
Var^{\ast }\left( S_{n}^{\ast }\right) =\sum_{A\in \mathcal{N}%
_{J}}\sum_{A^{\prime }\in \mathcal{N}_{J}}\mu _{A}\mu _{A^{\prime
}}Cov^{\ast }(\psi _{n,A}^{\ast },\psi _{n,A^{\prime }}^{\ast }),
\end{equation*}%
where $\psi _{n,A}^{\ast }\equiv h^{-d/2}(\zeta _{N,A}^{\ast }-\mathbf{E}%
^{\ast }\zeta _{N,A}^{\ast })$. By Lemma C6, we find that for $A,A\in 
\mathcal{N}_{J},$ 
\begin{equation*}
Cov^{\ast }(\psi _{n,A}^{\ast },\psi _{n,A^{\prime }}^{\ast })=\sigma
_{n,A,A^{\prime }}(B_{n,A}(c_{n};\mathcal{C}),B_{n,A^{\prime }}(c_{n};%
\mathcal{C}))+o_{P}(1),
\end{equation*}%
uniformly in $P\in \mathcal{P}$, yielding the desired result of (\ref{varvar}%
).

Combining Steps 1 and 2, we deduce that for some $C>0$,\textbf{\ }%
\begin{equation*}
\sup_{P\in \mathcal{P}}\left\vert Cov^{\ast }(S_{n}^{\ast },U_{n}^{\ast
})\right\vert \leq \sqrt{\bar{\varepsilon}}\cdot O_{P}(1)+o_{P}(h^{d/2}).
\end{equation*}%
Let $\tilde{\sigma}_{1}^{2}\equiv Var^{\ast }(S_{n}^{\ast })$ and $\tilde{%
\sigma}_{2}^{2}\equiv 1-\tilde{\alpha}_{P}$, where $\tilde{\alpha}_{P}\equiv
P^{\ast }\left\{ X_{i}^{\ast }\in \mathbf{R}^{d}\backslash \mathcal{C}%
\right\} $. Observe that%
\begin{equation*}
\tilde{\sigma}_{1}^{2}=\sigma _{n}(\mathcal{C})+o_{P}(1)>C_{1}+o_{P}(1),\ 
\mathcal{P}\text{-uniformly,}
\end{equation*}%
for some $C_{1}>0$ that does not depend on $n$ or $P$ by the assumption of
the lemma. Also note that 
\begin{equation*}
\tilde{\alpha}_{P}=\alpha _{P}+o_{P}(1)<1-C_{2}+o_{P}(1),\ \mathcal{P}\text{%
-uniformly,}
\end{equation*}%
for some $C_{2}>0$. Therefore, following the same arguments as in (\ref%
{dev67}), we obtain the desired result.\medskip

\noindent \textbf{Proof of Step 4:} We take $\{R_{n,\mathbf{i}}:\mathbf{i}%
\in \mathbb{Z}^{d}\}$, and define%
\begin{eqnarray*}
B_{A,x}(c_{n}) &\equiv &\left\{ \tau \in \mathcal{T}:(x,\tau )\in
B_{A}(c_{n})\right\} , \\
B_{n,\mathbf{i}} &\equiv &R_{n,\mathbf{i}}\cap \mathcal{C}\text{,} \\
B_{n,A,\mathbf{i}}(c_{n}) &\equiv &(B_{n,\mathbf{i}}\times \mathcal{T})\cap
B_{A}(c_{n}),
\end{eqnarray*}%
and $\mathcal{I}_{n}\equiv \{\mathbf{i}\in \mathbb{Z}_{n}^{d}:B_{n,\mathbf{i}%
}\neq \varnothing \}$ as in the proof of Step 4 in the proof of Lemma B8.
Also, define%
\begin{equation*}
\Delta _{n,A,\mathbf{i}}^{\ast }\equiv h^{-d/2}\int_{B_{n,\mathbf{i}%
}}\int_{B_{A,x}(c_{n})}\left\{ \Lambda _{A,p}(\mathbf{z}_{N,\tau }^{\ast
}(x))-\mathbf{E}^{\ast }\left[ \Lambda _{A,p}(\mathbf{z}_{N,\tau }^{\ast
}(x))\right] \right\} d\tau dx.
\end{equation*}%
Also, define%
\begin{eqnarray*}
\alpha _{n,\mathbf{i}}^{\ast } &\equiv &\frac{\sum_{A\in \mathcal{N}_{J}}\mu
_{A}\Delta _{n,A,\mathbf{i}}^{\ast }}{\sqrt{Var^{\ast }\left( S_{n}^{\ast
}\right) }}\mathbf{\ }\text{and} \\
u_{n,\mathbf{i}}^{\ast } &\equiv &\frac{1}{\sqrt{n}}\left\{
\sum_{i=1}^{N}1\left\{ X_{i}^{\ast }\in B_{n,\mathbf{i}}\right\} -nP^{\ast
}\{X_{i}^{\ast }\in B_{n,\mathbf{i}}\}\right\}
\end{eqnarray*}%
and write%
\begin{equation*}
\frac{S_{n}^{\ast }}{\sqrt{Var^{\ast }\left( S_{n}^{\ast }\right) }}=\sum_{%
\mathbf{i}\in \mathcal{I}_{n}}\alpha _{n,\mathbf{i}}^{\ast }\text{ and }%
U_{n}^{\ast }=\sum_{\mathbf{i}\in \mathcal{I}_{n}}u_{n,\mathbf{i}}^{\ast }.
\end{equation*}%
By the properties of Poisson processes, one can see that the array $%
\{(\alpha _{n,\mathbf{i}}^{\ast },u_{n,\mathbf{i}}^{\ast })\}_{\mathbf{i}\in 
\mathcal{I}_{n}}$ is an array of $1$-dependent random field under $P^{\ast }$%
. For any $q=(q_{1},q_{2})\in \mathbf{R}^{2}\backslash \{0\}$, let $y_{n,%
\mathbf{i}}^{\ast }\equiv q_{1}\alpha _{n,\mathbf{i}}^{\ast }+q_{2}u_{n,%
\mathbf{i}}^{\ast }$ and write%
\begin{equation*}
Var^{\ast }\left( \sum_{\mathbf{i}\in \mathcal{I}_{n}}y_{n,\mathbf{i}}^{\ast
}\right) =q_{1}^{2}+q_{2}^{2}(1-\tilde{\alpha}_{P})+2q_{1}q_{2}\tilde{c}%
_{n,P},
\end{equation*}%
uniformly over $P\in \mathcal{P}$, where $\tilde{c}_{n,P}=Cov^{\ast
}(S_{n}^{\ast },U_{n}^{\ast })$. On the other hand, following the proof of
Lemma A8 of \citeasnoun{LSW} using Lemma C4, we deduce that%
\begin{equation}
\sum_{\mathbf{i}\in \mathcal{I}_{n}}\mathbf{E}^{\ast }|y_{n,\mathbf{i}%
}^{\ast }|^{r}=o_{P}(1),\ \mathcal{P}\text{-uniformly,}  \label{conv_y2}
\end{equation}%
as $n\rightarrow \infty $, for any $r\in (2,(2p+2)/p]$, uniformly over $P\in 
\mathcal{P}$. By Theorem 1 of \citeasnoun{Shergin:93}, we have%
\begin{eqnarray*}
&&\sup_{t\in \mathbf{R}}\left\vert P^{\ast }\left\{ \frac{1}{\sqrt{%
q_{1}^{2}+q_{2}^{2}(1-\tilde{\alpha}_{P})+2q_{1}q_{2}\tilde{c}_{n,P}}}\sum_{%
\mathbf{i}\in \mathcal{I}_{n}}y_{n,\mathbf{i}}^{\ast }\leq t\right\} -\Phi(t) \right\vert \\
&\leq &\frac{C}{\left\{ q_{1}^{2}+q_{2}^{2}(1-\tilde{\alpha}_{P})+2q_{1}q_{2}%
\tilde{c}_{n,P}\right\} ^{r/2}}\left\{ \sum_{\mathbf{i}\in \mathcal{I}_{n}}%
\mathbf{E}^{\ast }|y_{n,\mathbf{i}}^{\ast }|^{r}\right\} ^{1/2}=o_{P}(1),
\end{eqnarray*}%
for some $C>0$ uniformly in $P\in \mathcal{P}$, by (\ref{conv_y2}). By Lemma
B2(i), we have for each $t\in \mathbf{R}$ and $q\in \mathbf{R}^{2}\backslash
\{\mathbf{0}\}$ as $n\rightarrow \infty ,$%
\begin{equation*}
\left\vert \mathbf{E}^{\ast }\left[ \exp \left( it\frac{q^{\top }H_{n}^{\ast
}}{\sqrt{q^{\top }\tilde{C}_{n}q}}\right) \right] -\exp \left( -\frac{t^{2}}{%
2}\right) \right\vert =o_{P}(1),
\end{equation*}%
uniformly in $P\in \mathcal{P}.$ Thus by Lemma B2(ii), for each $t\in 
\mathbf{R}^{2},$ we have%
\begin{equation*}
\left\vert P^{\ast }\left\{ \tilde{C}_{n}^{-1/2}H_{n}^{\ast }\leq t\right\}
-P\left\{ \mathbb{Z}\leq t\right\} \right\vert =o_{P}(1).
\end{equation*}%
Since the limit distribution of $\tilde{C}_{n}^{-1/2}H_{n}^{\ast }$ is
continuous, the convergence above is uniform in $t\in \mathbf{R}^{2}$.

\noindent (ii) We fix $P\in \mathcal{P}$ such that limsup$_{n\rightarrow
\infty }\sigma _{n}^{2}(\mathcal{C})=0.$ Then by (\ref{varvar}) above and
Lemma C6,%
\begin{equation*}
Var^{\ast }\left( S_{n}^{\ast }\right) =\sigma _{n}^{2}(\mathcal{C}%
)+o_{P}(1)=o_{P}(1).
\end{equation*}%
Hence, we find that $S_{n}^{\ast }=o_{P^{\ast }}(1)$ in $P$. The desired
result follows by applying Theorem 1 of \citeasnoun{Shergin:93} to the sum $%
U_{n}^{\ast }=\sum_{\mathbf{i}\in \mathcal{I}_{n}}u_{n,\mathbf{i}}^{\ast }$,
and then applying Lemma B2.
\end{proof}

\begin{LemmaD}
\label{lem-d9} \textit{Let }$\mathcal{C}$\textit{\ be the Borel set in Lemma
C8}.

\noindent \textit{(i) Suppose that the conditions of Lemma C8(i) are
satisfied. Then for each }$a>0$, \textit{as }$n\rightarrow \infty ,$%
\begin{equation*}
\sup_{P\in \mathcal{P}}P\left\{ \sup_{t\in \mathbf{R}}\left\vert P\left\{ 
\frac{h^{-d/2}\sum_{A\in \mathcal{N}_{J}}\mu _{A}\left\{ \zeta _{n,A}^{\ast
}-\mathbf{E}^{\ast }\zeta _{N,A}^{\ast }\right\} }{\sigma _{n}(\mathcal{C})}%
\leq t\right\} -\Phi (t)\right\vert >a\right\} \rightarrow 0.
\end{equation*}%
\noindent \textit{(ii) Suppose that the conditions of Lemma C8(ii) are
satisfied. Then for each }$a>0,$ \textit{as }$n\rightarrow \infty ,$%
\begin{equation*}
\sup_{P\in \mathcal{P}}P\left\{ \left\vert h^{-d/2}\sum_{A\in \mathcal{N}%
_{J}}\mu _{A}\left\{ \zeta _{n,A}^{\ast }-\mathbf{E}^{\ast }\zeta
_{N,A}^{\ast }\right\} \right\vert >a\right\} \rightarrow 0.
\end{equation*}
\end{LemmaD}

\begin{proof}[Proof of Lemma C\protect\ref{lem-d9}]
The proofs are precisely the same as those of Lemma B9, except that we use
Lemma C8 instead of Lemma B8 here.
\end{proof}

\begin{LemmaD}
\label{lem-d10} \textit{Suppose that the conditions of Lemma B5 hold. Then
for any small }$\nu >0,$\textit{\ there exists a positive sequence }$%
\varepsilon _{n}=o(h^{d})$\textit{\ such that for all }$r\in \lbrack 2,M/2]$%
\textit{\ (with }$M\ge 4$\textit{\ being as in Assumption A6(i)),}%
\begin{equation*}
\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\mathbf{E}||\Sigma
_{n,\tau ,\varepsilon _{n}}^{-1/2}(x)q_{n,\tau }(x;\eta _{n})||^{r}=O\left(
h^{-(r-2)\left( \frac{M-1}{M-2}\right) d-\nu }\right) ,
\end{equation*}%
\textit{where }$\eta _{n}\in \mathbf{R}^{J}$ \textit{is distributed as} $%
N(0,\varepsilon _{n}I_{J})$\textit{\ and independent of }$((Y_{i}^{\top
},X_{i}^{\top })_{i=1}^{\infty },N)$\textit{\ in the definition of }$%
q_{n,\tau }(x)$,\textit{\ and} 
\begin{equation}
\Sigma _{n,\tau ,\varepsilon _{n}}(x)\equiv \Sigma _{n,\tau ,\tau
}(x,0)+\varepsilon _{n}I_{J}\text{ and }q_{n,\tau }(x;\eta _{n})\equiv
q_{n,\tau }(x)+\eta _{n}\text{.}  \label{SIGMA}
\end{equation}%
\textit{Suppose furthermore that }$\lambda _{\min }(\Sigma _{n,\tau ,\tau
}(x,0))>c>0$\textit{\ for some }$c>0$\textit{\ that does not depend on }$n$%
\textit{\ or }$P\in \mathcal{P}$.\textit{\ Then}%
\begin{equation*}
\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\mathbf{E}||\Sigma
_{n,\tau ,\varepsilon _{n}}^{-1/2}(x)q_{n,\tau }(x;\eta _{n})||^{r}=O\left(
h^{-(r-2)d/2}\right) .
\end{equation*}
\end{LemmaD}

\begin{proof}[Proof of Lemma C\protect\ref{lem-d10}]
We first establish the following fact.

\noindent \textbf{Fact:} Suppose that $W$ is a random vector such that $%
\mathbf{E}||W||^{2}\leq c_{W}$ for some constant $c_{W}>0$. Then, for any $%
r\geq 2$ and a positive integer $m\geq 1,$ 
\begin{equation*}
\mathbf{E}\left[ ||W||^{r}\right] \leq C_{m}\left( \mathbf{E}\left[
||W||^{a_{m}(r)}\right] \right) ^{1/(2^{m})},
\end{equation*}%
where $a_{m}(r)=2^{m}(r-2)+2$, and $C_{m}>0$ is a constant that depends only
on $m$ and $c_{W}$.\medskip \newline
\textbf{Proof of Fact:}\ The result follows by repeated application of
Cauchy-Schwarz inequality:%
\begin{equation*}
\mathbf{E}||W||^{r}\leq \left( \mathbf{E}||W||^{2(r-1)}\right) ^{1/2}\left( 
\mathbf{E}||W||^{2}\right) ^{1/2}\leq c_{W}^{1/2}\left( \mathbf{E}%
||W||^{2(r-1)}\right) ^{1/2},
\end{equation*}%
where we replace $r$ on the left hand side by $2(r-1)$, and repeat the
procedure to obtain Fact.\medskip \newline
Let us consider the first statement of the lemma. Using Fact, we take a
small $\nu _{1}>0$ and $\varepsilon _{n}=h^{d+\nu _{1}}$, and choose a
largest integer $m\geq 1$ such that $a_{m}(r)\leq M$. Such an $m$ exists
because $2\leq r\leq M/2$. We bound%
\begin{equation*}
\mathbf{E}||\Sigma _{n,\tau ,\varepsilon _{n}}^{-1/2}(x)q_{n,\tau }(x;\eta
_{n})||^{r}\leq C_{m}\left( \mathbf{E}||\Sigma _{n,\tau ,\varepsilon
_{n}}^{-1/2}(x)q_{n,\tau }(x;\eta _{n})||^{a_{m}(r)}\right) ^{1/(2^{m})}.
\end{equation*}%
By Lemma B5, we find that 
\begin{eqnarray}
&&\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\mathbf{E}||\Sigma
_{n,\tau ,\varepsilon _{n}}^{-1/2}(x)q_{n,\tau }(x;\eta _{n})||^{a_{m}(r)}
\label{ineqs4} \\
&\leq &\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\lambda _{\max
}^{a_{m}(r)/2}\left( \Sigma _{n,\tau ,\varepsilon _{n}}^{-1}(x)\right) 
\mathbf{E}||q_{n,\tau }(x;\eta _{n})||^{a_{m}(r)}  \notag \\
&\leq &\lambda _{\min }^{-a_{m}(r)/2}\left( \varepsilon _{n}I_{J}\right)
h^{(1-(a_{m}(r)/2))d}.  \notag
\end{eqnarray}%
By the definition of $\varepsilon _{n}=h^{d+\nu _{1}}$,%
\begin{equation*}
\varepsilon
_{n}^{-a_{m}(r)/2}h^{(1-(a_{m}(r)/2))d}=h^{(1-a_{m}(r))d-a_{m}(r)\nu _{1}/2}.
\end{equation*}%
We conclude that 
\begin{eqnarray*}
\mathbf{E}||\Sigma _{n,\tau ,\varepsilon _{n}}^{-1/2}(x)q_{n,\tau }(x;\eta
_{n})||^{r} &\leq &C_{m}\left( h^{(1-a_{m}(r))d-a_{m}(r)\nu _{1}/2}\right)
^{1/2^{m}} \\
&=&C_{m}\left( h^{(-1-2^{m}(r-2))d-(2^{m}(r-2)+2)\nu _{1}/2}\right)
^{1/2^{m}} \\
&=&C_{m}h^{(-2^{-m}-(r-2))d-((r-2)+2^{-m+1})\nu _{1}/2}.
\end{eqnarray*}%
Since $a_{m}(r)\leq M$, or $2^{-m}\geq (r-2)/(M-2)$, the last term is
bounded by%
\begin{equation*}
C_{m}h^{-(r-2)\left( \frac{M-1}{M-2}\right) d-\left( (r-2)+\frac{2(r-2)}{M-2}%
\right) \nu _{1}/2}.
\end{equation*}%
By taking $\nu _{1}$ small enough, we obtain the desired result.

Now, let us turn to the second statement of the lemma. Since 
\begin{equation*}
\lambda _{\max }^{a_{m}(r)/2}\left( \Sigma _{n,\tau ,\varepsilon
_{n}}^{-1}(x)\right) <c^{-a_{m}(r)/2},
\end{equation*}%
the last bound in (\ref{ineqs4}) turns out to be 
\begin{equation*}
c^{-a_{m}(r)/2}h^{(1-(a_{m}(r)/2))d}.
\end{equation*}%
Therefore, we conclude that 
\begin{eqnarray*}
\mathbf{E}||\Sigma _{n,\tau ,\varepsilon _{n}}^{-1/2}(x)q_{n,\tau }(x;\eta
_{n})||^{r} &\leq &C_{m}\left( c^{-a_{m}(r)/2}h^{(1-(a_{m}(r)/2))d}\right)
^{1/2^{m}} \\
&=&C_{m}c^{-\{(r-2)+2^{1-m}\}/2}h^{(2^{-m}-\{(r-2)+2^{1-m}\}/2)d} \\
&=&C_{m}c^{-\{(r-2)+2^{1-m}\}/2}h^{-(r-2)d/2}.
\end{eqnarray*}%
Again, using the inequality $2^{-m}\geq (r-2)/(M-2)$, we obtain the desired
result.
\end{proof}

\begin{LemmaD}
\label{lem-d11} \textit{Suppose that the conditions of Lemma C5 hold. Then
for any small }$\nu >0,$\textit{\ there exists a positive sequence }$%
\varepsilon _{n}=o(h^{d})$\textit{\ such that for all }$r\in \lbrack 2,M/2]$%
\textit{\ (with }$M\ge 4$\textit{\ being as in Assumption A6(i)),}%
\begin{equation*}
\sup_{(x,\tau )\in \mathcal{S}}\mathbf{E}^{\ast }||\tilde{\Sigma}_{n,\tau
,\varepsilon _{n}}^{-1/2}(x)q_{n,\tau }^{\ast }(x;\eta
_{n})||^{r}=O_{P}\left( h^{-(r-2)\left( \frac{M-1}{M-2}\right) d-\nu
}\right) ,\text{\textit{\ uniformly in }}P\in \mathcal{P,}
\end{equation*}%
\textit{where }$\eta _{n}\in \mathbf{R}^{J}$ \textit{is distributed as} $%
N(0,\varepsilon _{n}I_{J})$\textit{\ and independent of }$((Y_{i}^{\ast \top
},X_{i}^{\ast \top })_{i=1}^{n},(Y_{i}^{\top },X_{i}^{\top })_{i=1}^{n},N)$%
\textit{\ in the definition of }$q_{n,\tau }^{\ast }(x)$,\textit{\ and}%
\begin{equation*}
\tilde{\Sigma}_{n,\tau ,\varepsilon _{n}}(x)\equiv \tilde{\Sigma}_{n,\tau
,\tau }(x,0)+\varepsilon _{n}I_{J}.
\end{equation*}%
\textit{Suppose furthermore that }%
\begin{equation*}
\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}P\left\{ \lambda
_{\min }(\tilde{\Sigma}_{n,\tau ,\tau }(x,0))>c\right\} \rightarrow 0,
\end{equation*}%
\textit{for some }$c>0$\textit{\ that does not depend on }$n$\textit{\ or }$%
P\in \mathcal{P}$.\textit{\ Then}%
\begin{equation*}
\sup_{(x,\tau )\in \mathcal{S}}\mathbf{E}^{\ast }||\tilde{\Sigma}_{n,\tau
,\varepsilon _{n}}^{-1/2}(x)q_{n,\tau }^{\ast }(x;\eta
_{n})||^{r}=O_{P}\left( h^{-(r-2)d/2}\right) ,\text{\textit{\ uniformly in }}%
P\in \mathcal{P.}
\end{equation*}
\end{LemmaD}

\begin{proof}[Proof of Lemma C\protect\ref{lem-d11}]
The proof is precisely the same as that of Lemma C10, where we use Lemma C5
instead of Lemma B5.
\end{proof}

We let for a sequence of Borel sets $B_{n}$ in $\mathcal{S}$ and $\lambda
\in \{0,d/4,d/2\}$, $A\subset \mathbb{N}_{J}$, and a fixed bounded function $%
\delta $ on $\mathcal{S}$, 
\begin{eqnarray*}
a_{n}^{R}(B_{n}) &\equiv &\int_{B_{n}}\mathbf{E}\left[ \Lambda _{A,p}(\sqrt{%
nh^{d}}\mathbf{z}_{N,\tau }(x)+h^{\lambda }\delta (x,\tau ))\right]
dQ(x,\tau )\text{ } \\
a_{n}^{R\ast }(B_{n}) &\equiv &\int_{B_{n}}\mathbf{E}^{\ast }\left[ \Lambda
_{A,p}(\sqrt{nh^{d}}\mathbf{z}_{N,\tau }^{\ast }(x)+h^{\lambda }\delta
(x,\tau ))\right] dQ(x,\tau ),\text{ and} \\
a_{n}(B_{n}) &\equiv &\int_{B_{n}}\mathbf{E}\left[ \Lambda _{A,p}(\mathbb{W}%
_{n,\tau ,\tau }^{(1)}(x,0)+h^{\lambda }\delta (x,\tau ))\right] dQ(x,\tau ),
\end{eqnarray*}%
where $\mathbf{z}_{N,\tau }^{\ast }(x)$ is a random vector whose $j$-th
entry is given by%
\begin{equation*}
z_{N,\tau ,j}^{\ast }(x)\equiv \frac{1}{nh^{d}}\sum_{i=1}^{N}\beta
_{n,x,\tau ,j}(Y_{ij}^{\ast },(X_{i}^{\ast }-x)/h)-\frac{1}{h^{d}}\mathbf{E}%
^{\ast }\left[ \beta _{n,x,\tau ,j}(Y_{ij}^{\ast },(X_{i}^{\ast }-x)/h)%
\right] \text{.}
\end{equation*}

\begin{LemmaD}
\label{lem-d12} \textit{Suppose that the conditions of Lemmas C10 and C11
hold and that }%
\begin{equation*}
n^{-1/2}h^{-\left( \frac{3M-4}{2M-4}\right) d-\nu }\rightarrow 0,
\end{equation*}%
\textit{\ as }$n\rightarrow \infty $, \textit{for some small }$\nu >0$. 
\textit{Then for any sequence of Borel sets }$B_{n}$ \textit{in} $\mathcal{S}
$\textit{,}%
\begin{eqnarray*}
\sup_{P\in \mathcal{P}}\left\vert a_{n}^{R}(B_{n})-a_{n}(B_{n})\right\vert
&=&o(h^{d/2})\text{ \textit{and}} \\
\sup_{P\in \mathcal{P}}P\left\{ \left\vert a_{n}^{R\ast
}(B_{n})-a_{n}(B_{n})\right\vert >ah^{d/2}\right\} &=&o(1).
\end{eqnarray*}
\end{LemmaD}

\begin{proof}[Proof of Lemma C\protect\ref{lem-d12}]
For the statement, it suffices to show that uniformly in $P\in \mathcal{P}$,%
\begin{eqnarray}
\sup_{(x,\tau )\in \mathcal{S}}\left\vert 
\begin{array}{c}
\mathbf{E}\Lambda _{A,p}(\sqrt{nh^{d}}\mathbf{z}_{N,\tau }(x)+h^{\lambda
}\delta (x,\tau )) \\ 
-\mathbf{E}\Lambda _{A,p}(\mathbb{W}_{n,\tau ,\tau }^{(1)}(x,0)+h^{\lambda
}\delta (x,\tau ))%
\end{array}%
\right\vert &=&o(h^{d/2})\text{, and}  \label{convs2} \\
\sup_{(x,\tau )\in \mathcal{S}}\left\vert 
\begin{array}{c}
\mathbf{E}^{\ast }\Lambda _{A,p}(\sqrt{nh^{d}}\mathbf{z}_{N,\tau }^{\ast
}(x)+h^{\lambda }\delta (x,\tau )) \\ 
-\mathbf{E}\Lambda _{A,p}(\mathbb{W}_{n,\tau ,\tau }^{(1)}(x,0)+h^{\lambda
}\delta (x,\tau ))%
\end{array}%
\right\vert &=&o_{P}(h^{d/2}).  \notag
\end{eqnarray}%
We prove the first statement of (\ref%
{convs2}). The proof of the second statement of (\ref{convs2}) can be done
in a similar way.

Take small $\nu >0.$ We apply Lemma C10 by choosing a positive sequence $%
\varepsilon _{n}=o(h^{d})$ such that for any $r\in \lbrack 2,M/2],$%
\begin{equation}
\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\mathbf{E}||\Sigma
_{n,\tau ,\varepsilon _{n}}^{-1/2}(x)q_{n,\tau }(x;\eta _{n})||^{r}=O\left(
h^{-(r-2)\left( \frac{M-1}{M-2}\right) d-\nu }\right) ,  \label{rate}
\end{equation}%
where $q_{n,\tau }(x;\eta _{n})$ and $\Sigma _{n,\tau ,\varepsilon _{n}}(x)$
are as in Lemma C10. We follow the arguments in the proof of Step 2 in Lemma
B6 to bound the left-hand side in the first supremum in (\ref{convs2}) by 
\begin{equation*}
\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\left\vert \mathbf{E}%
\Lambda _{A,p}(\sqrt{nh^{d}}\mathbf{z}_{N,\tau }(x;\eta _{n})+h^{\lambda
}\delta (x,\tau ))-\mathbf{E}\Lambda _{A,p}(\mathbb{W}_{n,\tau ,\tau
,\varepsilon _{n}}^{(1)}(x,0)+h^{\lambda }\delta (x,\tau ))\right\vert +C%
\sqrt{\varepsilon _{n}},
\end{equation*}%
for some $C>0$, where\ 
\begin{equation*}
\mathbf{z}_{N,\tau }(x;\eta _{n})\equiv \mathbf{z}_{N,\tau }(x)+\eta _{n}/%
\sqrt{nh^{d}},
\end{equation*}%
and $\mathbb{W}_{n,\tau ,\tau ,\varepsilon _{n}}^{(1)}(x,0)$ is as defined
in (\ref{Ws}). Let%
\begin{eqnarray*}
\mathbf{\xi }_{N,\tau }(x;\eta _{n}) &\equiv &\sqrt{nh^{d}}\Sigma _{n,\tau
,\varepsilon _{n}}^{-1/2}(x)\cdot \mathbf{z}_{N,\tau }(x;\eta _{n})\text{ and%
} \\
\mathbb{Z}_{n,\tau ,\tau ,\varepsilon _{n}}^{(1)}(x,0) &\equiv &\Sigma
_{n,\tau ,\varepsilon _{n}}^{-1/2}(x)\cdot \mathbb{W}_{n,\tau ,\tau
,\varepsilon _{n}}^{(1)}(x,0).
\end{eqnarray*}%
We rewrite the previous absolute value as%
\begin{equation}
\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\left\vert \mathbf{E}%
\Lambda _{A,n,p}^{\Sigma }(\sqrt{nh^{d}}\mathbf{\xi }_{N,\tau }(x;\eta
_{n}))-\mathbf{E}\Lambda _{n,p}^{\Sigma }(\mathbb{Z}_{n,\tau ,\tau
,\varepsilon _{n}}^{(1)}(x,0))\right\vert ,  \label{bdd4}
\end{equation}%
where $\Lambda _{A,n,p}^{\Sigma }(\mathbf{v})\equiv \Lambda _{A,p}(\Sigma
_{n,\tau ,\varepsilon _{n}}^{1/2}(x)\mathbf{v}+h^{\lambda }\delta (x,\tau ))$%
. Note that the condition for $M$ in Assumption A6(i) that $M\geq 2(p+2)$,
we can choose $r=\max \{p,3\}$. Then $r\in \lbrack 2,M/2]$ as required.
Using Theorem 1 of \citeasnoun{Sweeting:77}, we bound the above supremum by
(with $r=\max \{p,3\}$)%
\begin{eqnarray*}
&&\frac{C_{1}}{\sqrt{n}}\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}%
}\mathbf{E}||\Sigma _{n,\tau ,\varepsilon _{n}}^{-1/2}(x)q_{n,\tau }(x;\eta
_{n})||^{3} \\
&&+\frac{C_{2}}{\sqrt{n^{r-2}}}\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in 
\mathcal{P}}\mathbf{E}||\Sigma _{n,\tau ,\varepsilon
_{n}}^{-1/2}(x)q_{n,\tau }(x;\eta _{n})||^{r} \\
&&+C_{3}\sup_{(x,\tau )\in \mathcal{S}}\sup_{P\in \mathcal{P}}\mathbf{E}%
\omega _{n,p}\left( \mathbb{Z}_{n,\tau ,\tau ,\varepsilon _{n}}^{(1)}(x,0);%
\frac{C_{4}}{\sqrt{n}}\mathbf{E}||\Sigma _{n,\tau ,\varepsilon
_{n}}^{-1/2}(x)q_{n,\tau }(x;\eta _{n})||^{3}\right) ,
\end{eqnarray*}%
for some positive constants $C_{1},C_{2},C_{3},$ and $C_{4}$, where%
\begin{equation*}
\omega _{n,p}\left( \mathbf{v};c\right) \equiv \sup \left\{ |\Lambda
_{A,n,p}^{\Sigma }(\mathbf{v})-\Lambda _{A,n,p}^{\Sigma }(\mathbf{y})|:%
\mathbf{y}\in \mathbf{R}^{|A|},||\mathbf{v}-\mathbf{y}||\leq c\right\} .
\end{equation*}%
The proof is complete by (\ref{rate}) and by the condition $%
n^{-1/2}h^{-\left( \frac{3M-4}{2M-4}\right) d-\nu }\rightarrow 0$.
\end{proof}

%\clearpage

\section{Proof of Theorem AUC\ref{thm:AUC1}}\label{sec:appendix-D}

The conclusion of Theorem AUC\ref{thm:AUC1} follows immediately from 
Theorem 1, provided that all the regularity conditions in Theorem 1 are satisfied.
The following lemma shows that Assumptions AUC1-AUC4 are sufficient conditions for that purpose. 
One key condition to check the regularity condition of Theorem 1 is to establish 
asymptotic linear representations in
Assumptions A1 and B1. We borrow the results from  \citeasnoun{LSW-quantile:15}.


\begin{LemmaAUC}
Suppose that Assumptions AUC1-AUC4 hold. Then Assumptions A1-A6
and B1-B4 hold with the following definitions: $%
J=2,\ r_{n,j}\equiv \sqrt{nh^{d}},$%
\begin{eqnarray*}
v_{n,\tau ,1}(x) &\equiv &\mathbf{e}_{1}^{\top }\{\gamma _{\tau
,2}(x)-\gamma _{\tau ,3}(x)\}, \\
v_{n,\tau ,2}(x) &\equiv &\underline{b}-\mathbf{e}_{1}^{\top }\{2\gamma
_{\tau ,2}(x)-\gamma _{\tau ,3}(x)\}, \\
\beta _{n,x,\tau ,1}(Y_{i},z) &\equiv &\alpha _{n,x,\tau ,2}(Y_{i},z)-\alpha
_{n,x,\tau ,3}(Y_{i},z),\text{ \textit{and}} \\
\beta _{n,x,\tau ,2}(Y_{i},z) &\equiv &-2\alpha _{n,x,\tau
,2}(Y_{i},z)+\alpha _{n,x,\tau ,3}(Y_{i},z)\text{\textit{,}}
\end{eqnarray*}%
\textit{where }$\tilde{l}_{\tau }(u)\equiv \tau -1\{u\leq 0\},$ $%
Y_{i}=\{(B_{\ell i},L_{i}):\ell =1,\ldots ,L_{i}\}$\textit{, and}%
\begin{equation*}
\alpha _{n,x,\tau ,k}(Y_{i},z)\equiv -1\left\{ L_{i}=k\right\} \sum_{l=1}^{k}%
\tilde{l}_{\tau }\left( B_{\ell i}-\gamma _{\tau ,k}^{\top }(x)\cdot H\cdot
c\left( z\right) \right) \mathbf{e}_{1}^{\top }M_{n,\tau ,k}^{-1}(x)c\left(
z\right) K\left( z\right) \text{.}
\end{equation*}
\end{LemmaAUC}

\begin{proof}[Proof of Lemma AUC1]
First, let us turn to Assumption A1. By Assumptions AUC2 and AUC3, it
suffices to consider $\hat{v}_{\tau ,2}(x)$ that uses $\underline{b}$
instead of $\underline{\hat{b}}$. The asymptotic linear representation in
Assumption A1 follows from Theorem 1 of \citeasnoun{LSW-quantile:15}. The error rate $o_{P}(\sqrt{%
h^{d}})$ in Assumption A1 is satisfied, because 
\begin{equation}
h^{-d/2}\left( \frac{\log ^{1/2}n}{n^{1/4}h^{d/4}}\right)
=n^{-1/4}h^{-3d/4}\log ^{1/2}n\rightarrow 0,  \label{error}
\end{equation}%
by Assumption AUC2(ii) and the condition $r>3d/2-1$. Assumption A2 follows
because both $\beta _{n,x,\tau ,1}(Y_{i},z)$ and $\beta _{n,x,\tau
,2}(Y_{i},z)$ have a multiplicative component of $K(z)$ which has a compact
support by Assumption AUC2(i). As for Assumption A3, we use Lemma 2. First define 
\begin{eqnarray*}
e_{x,\tau ,k,li} &\equiv &1\left\{ L_{i}=k\right\} \tilde{l}_{\tau }\left(
B_{li}-\gamma _{\tau ,k}^{\top }(x)\cdot H\cdot c\left( \frac{X_{i}-x}{h}%
\right) \right) \text{ and} \\
\xi _{x,\tau ,k,i} &\equiv &\mathbf{e}_{1}^{\top }M_{n,\tau
,k}^{-1}(x)c\left( \frac{X_{i}-x}{h}\right) K\left( \frac{X_{i}-x}{h}\right)
\end{eqnarray*}%
First observe that for each fixed $x_{2}\in \mathbf{R}^{d},\tau _{2}\in 
\mathcal{T}$, and $\lambda >0$,%
\begin{eqnarray}
&&\mathbf{E}\left[ \sup_{||x_{2}-x_{3}||+||\tau _{2}-\tau _{3}||\leq \lambda
}\left( \alpha _{n,x_{2},\tau _{2},2}\left( Y_{i,}\frac{X_{i}-x_{2}}{h}%
\right) -\alpha _{n,x_{3},\tau _{3},2}\left( Y_{i,}\frac{X_{i}-x_{3}}{h}%
\right) \right) ^{2}\right]  \label{develop} \\
&\leq &2\sum_{l=1}^{k}\mathbf{E}\left[ \mathbf{E}\left[
\sup_{||x_{2}-x_{3}||+||\tau _{2}-\tau _{3}||\leq \lambda }\left(
e_{x_{2},\tau _{2},k,li}-e_{x_{3},\tau _{3},k,li}\right) ^{2}|X_{i}\right]
\xi _{x_{2},\tau _{2},k,i}^{2}\right]  \notag \\
&&+2\sum_{l=1}^{k}\mathbf{E}\left[ \sup_{||x_{2}-x_{3}||+||\tau _{2}-\tau
_{3}||\leq \lambda }\left( \xi _{x_{2},\tau _{2},k,i}-\xi _{x_{3},\tau
_{3},k,i}\right) ^{2}\right] .  \notag
\end{eqnarray}%
Using Lipschitz continuity of the conditional density of $B_{li}$ given $%
L_{i}=k$ and $X_{i}=x$ in $(x,\tau )$ and Lipschitz continuity of $\gamma
_{\tau ,k}(x)$ in $(x,\tau )$ (Assumption AUC1), we find that the first term
is bounded by $Ch^{-s_{1}}\lambda $ for some $C>0$ and $s_{1}>0$. Since 
\begin{equation*}
M_{n,\tau ,k}(x)=kP\left\{ L_{i}=k|X_{i}=x\right\} f_{\tau ,k}(0|x)f(x)\int
K(t)c(t)c(t)^{\top }dt+o(1),
\end{equation*}%
we find that $M_{n,\tau ,k}^{-1}(x)$ is Lipschitz continuous in $(x,\tau )$
by Assumptions AUC1. Hence the last term in (\ref{develop}) is also bounded
by $Ch^{-s_{2}}\lambda ^{2}$ for some $C>0$ and $s_{2}>0$. Therefore, if we
take%
\begin{equation*}
b_{n,ij}(x,\tau )=\alpha _{n,x,\tau ,2}\left( Y_{i,}\frac{X_{i}-x}{h}\right)
,
\end{equation*}%
this function satisfies the condition in Lemma 2. Also, observe that%
\begin{equation*}
\mathbf{E}\left[ \left\vert \alpha _{n,x,\tau ,2}\left( Y_{i,}\frac{X_{i}-x}{%
h}\right) \right\vert ^{4}\right] \leq C,
\end{equation*}%
because $\alpha _{n,x,\tau ,2}(\cdot ,\cdot )$ is uniformly bounded. We also
obtain the same result for $\alpha _{n,x,\tau ,3}(\cdot ,\cdot )$. Thus the
conditions of Lemma 2 are satisfied with $b_{n,ij}(x,\tau )$ taken to
be $\beta _{n,x,\tau ,1}(Y_{i},(X_{i}-x)/h)$ or $\beta _{n,x,\tau
,2}(Y_{i},(X_{i}-x)/h)$. Now Assumption A3 follows from Lemma 2(i). The rate
condition in Assumption A4(i) is satisfied by Assumption AUC2(ii).
Assumption A4(ii) is imposed directly by Assumption AUC4(i). Since we are taking $\hat{\sigma}_{\tau ,j}(x)=%
\hat{\sigma}_{\tau ,j}^{\ast }(x)=1$, it suffices to take $\sigma _{n,\tau
,j}(x)=1$ in Assumption A5 and Assumption B3.$\ $Assumption A6(i) is
satisfied because $\beta _{n,x,\tau ,j}$ is bounded. 
Assumption A6(ii) is imposed directly by Assumption AUC4(ii).
Assumption B1 follows
by Lemma QR2  of \citeasnoun{LSW-quantile:15}. Assumption B2 follows from Lemma 2(ii). Assumption B4
follows from the rate condition in Assumption AUC2(ii). In fact, when $\beta
_{n,x,\tau ,j}$ is bounded, the rate condition in Assumption B4 is reduced
to $n^{-1/2}h^{-3d/2-\nu }\rightarrow 0$, as $n\rightarrow \infty $, for
some small number $\nu >0$.
\end{proof}

\section{Potential Areas of Applications}\label{sec:app:literature}


Econometric models of games belong to a related but distinct branch of the
literature, compared to the auction models. In this literature, inference on
many game theoretic models are recently based on partial identification or
functional inequalities. For example, see \citeasnoun{Tamer:03}, %
\citeasnoun{Andrews/Berry/Jia:04}, \citeasnoun{Berry/Tamer:07}, %
\citeasnoun{Aradillas-Lopez/Tamer:08}, \citeasnoun{Ciliberto/Tamer:09}, %
\citeasnoun{Beresteanu/Molchanov/Molinari:11}, \citeasnoun{Galichon/Henry:11}%
, \citeasnoun{Cheser/Rosen:12}, and \citeasnoun{Aradillas-Lopez/Rosen:13},
among others. See \citeasnoun{dePaula:13} and references therein for a broad
recent development in this literature. Our general method provides
researchers in this field with a new inference tool when they have continuous
covariates.

Inequality restrictions also arise in testing revealed preferences. %
\citeasnoun{Blundell/Browning/Crawford:08} used revealed preference
inequalities to provide the nonparametric bounds on average consumer
responses to price changes. In addition, %
\citeasnoun{Blundell/Kristensen/Matzkin:11} used the same inequalities to
bound quantile demand functions. It would be possible to use our framework
to test revealed preference inequalities either in average demand functions
or in quantile demand functions. See also \citeasnoun{Hoderlein:Stoye:13}
and \citeasnoun{Kitamura:Stoye:13} for related issues of testing revealed
preference inequalities.

In addition to the literature mentioned above, many results on partial
identification can be written as functional inequalities. See, e.g., %
\citeasnoun{Imbens/Manski:04}, \citeasnoun{Manski:03}, \citeasnoun{Manski:07}%
, \citeasnoun{Manski/Pepper:00}, \citeasnoun{Tamer:10}, %
\citeasnoun{Cheser/Rosen:13}, and references therein.




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