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  pdftitle={On the standardization of the statistics in Bugni (2014)},
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\defcitealias{CHT07}{CHT}

\begin{document}
\sloppy

\title{Supplemental Material to ``Comparison of inferential methods in partially identified models in terms of error in coverage probability''\thanks{Thanks to an anonymous referee for suggesting me to explore this extension. Takuya Ura provided excellent research assistance. Any and all errors are my own.}}
\author{Federico A. Bugni\\Department of Economics\\Duke University\\ \href{mailto:federico.bugni@duke.edu}{\texttt{federico.bugni@duke.edu}.}}
\date{\today }
\maketitle

\begin{abstract}

This supplement provides all the necessary results to show that a standardized version of the statistics considered in \cite{bugni:2014} satisfy the same conclusions regarding the rate of convergence of the error in the coverage probability. To establish these findings, we require strengthening the finite moment requirements from finite fourth absolute moments to slightly over sixth absolute moments.

\vfill 

% \begin{description}
% 	\item \textsc{Keywords:} Partial Identification, Moment Inequalities, StandizationInference, Hypothesis Test, Bootstrap, Subsampling, Asymptotic Approximation, Rates of Convergence, Error in Coverage Probability.
% 	\item[]\textsc{JEL Classification:} C01, C12, C15. 
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\end{abstract}

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\section{Introduction}

\cite{bugni:2014} considers the problem of inference in a partially identified moment (in)equality models based on the criterion function approach. According to Assumptions CF.1-CF.2, the sample analogue criterion functions are of the following form
\begin{equation}
Q_{n}(\theta )=G( \{ [ \sqrt{n}\bar{m}_{n,j}(\theta )] _{-}\} _{j=1}^{J}) , \label{eq:CFsimple}
\end{equation}
where $G$ is a non-stochastic function. The fact that $G$ is non-stochastic implies that the sample moment conditions are not allowed to be standardized, i.e., divided by the sample standard deviation. This could be considered an important limitation relative to the existing literature. 

The objective of this supplement is to show that the results in \cite{bugni:2014} can be extended to allow for standardized sample moment conditions, i.e., criterion functions of the form
\begin{equation}
Q_{n}(\theta )=G( \{ [  \sqrt{n}\bar{m}_{n,j}(\theta )/\hat{ \sigma}_{n,j}(\theta )] _{-}\} _{j=1}^{J}) , \label{eq:CFstd}
\end{equation}
where $\hat{\sigma}_{n,j}^{2}(\theta )\equiv n^{-1}\sum_{i=1}^{n}( m(W_{i},\theta )-\bar{m}_{n,j}(\theta )) ^{2}$ is the sample variance of $\{m_{j}(W_{i},\theta )\}_{i=1}^{n}$ for $j=1,\dots ,J$.

The strategy to establish these results follows the one used in \cite{bugni:2014}. The results in that paper are based on several representation theorems that express a statistic of interest as a well-understood random variable plus an error term, which is shown to converge to zero at a sufficiently fast rate. These type of representation results are obtained for the sample statistic and each of the approximating statistics (bootstrap, asymptotic approximation, subsampling 1, and subsampling 2). This supplement shows that standardization adds a new component to the error term of each of these representation results. As a consequence, all of the results in \cite{bugni:2014} extend for the criterion functions in Eq.\ \eqref{eq:CFstd} as long as this additional component converges to zero at an appropriate rate. This supplement shows that occurs, provided that we strengthen the finite moment requirements from finite fourth absolute moments to slightly over sixth absolute moments.

We introduce the following notation in addition to the one used in \cite{bugni:2014}. For all $j=1,\dots ,J$,
\begin{eqnarray*}
\sigma _{j}^{2}(\theta ) &\equiv &V[m_{j}(Z,\theta )] \\
\hat{\sigma}_{n,j}^{2}(\theta ) &\equiv &n^{-1}\sum_{i=1}^{n}\left( m_{j}(Z_{i},\theta )-\bar{m}_{n,j}(\theta )\right) ^{2} \\
\tilde{\sigma}_{n,j}^{2}(\theta ) &\equiv &n^{-1}\sum_{i=1}^{n}(m_{j}(Z_{i},\theta )-E[m_{j}(Z_{i},\theta )])^{2} \\
v_{n,j}^{s}(\theta ) &\equiv &\sqrt{n}(\bar{m}_{n,j}(\theta )-E[m_{j}(Z,\theta )])/\sigma _{j}(\theta ), \\
\tilde{v}_{n,j}^{s}(\theta ) &\equiv &\sqrt{n}(\bar{m}_{n,j}(\theta )-E[m_{j}(Z,\theta )])/\hat{\sigma}_{n,j}(\theta ), \\
v_{n,j}^{s,\ast }(\theta ) &\equiv &\sqrt{n}\left( \bar{m}_{n,j}^{\ast }(\theta )-\bar{m}_{n,j}(\theta )\right) /\hat{\sigma}_{n,j}(\theta ),
\end{eqnarray*}
where the superscript ``$s$'' in $v_{n,j}^{s}(\theta )$, $\tilde{v} _{n,j}^{s}(\theta )$, and $v_{n,j}^{s,\ast }(\theta )$ denotes that the sample statistic is standardized.

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\section{Assumptions}

In order to develop the results in this supplement, we replace Assumptions A.5, CF.1, and CF.2 in \cite{bugni:2014} by the following alternative assumptions.

\begin{assumptionA} \label{ass:A5new} 
	$E[||m(Z,\theta )||^{6+\delta }]<\infty $ for some $\delta >0$.
\end{assumptionA}

\begin{assumptionCF} \label{ass:CF1new}
	The population criterion function is $Q(\theta )=G(\{[E[m_{j}(Z,\theta )]/\sigma _{j}(\theta )]_{-}\}_{j=1}^{J})$, where $G:\mathbb{R}_{+}^{J}\to  \mathbb{R}$ is a non-stochastic and non-negative function that is strictly increasing in every coordinate, weakly convex, continuous, homogeneous of degree $\beta>0$, and satisfies $G(y)=0$ if and only if $y={\bf 0}_{J}$.
\end{assumptionCF}

\begin{assumptionCF} \label{ass:CF2new}
	The population criterion function is $Q(\theta )=G(\{[E[m_{j}(Z,\theta )]/\sigma _{j}(\theta )]_{-}\}_{j=1}^{J})$, where $G:\mathbb{R}_{+}^{J}\to \mathbb{R}$ is one of the following two functions: (a) $ G(x)=\sum_{j=1}^{J}\varpi_{j}x_{j}$ or (b) $G(x)=\max \{\varpi_{j}x_{j}\}_{j=1}^{J}$, where $\varpi\in \mathbb{R}_{+}^{J}$ is an arbitrary vector of positive constants.
\end{assumptionCF}

Assumption \ref{ass:A5new} strengthens the finite moment requirements from finite fourth absolute moments in Assumption A.5 to slightly over sixth absolute moments. By virtue of Assumptions \ref{ass:CF1new}-\ref{ass:CF2new}, the properly scaled sample analogue criterion function is now given by Eq.\ \eqref{eq:CFstd}.

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\section{Representation results with standardization}

This section establishes representation results for the sample statistic and each of the approximating statistics. We note that all the proofs and several intermediate results are collected in the appendix of this supplement. We begin with the representation result for the standardized sample statistic.

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\begin{theorem}[Representation result - Sample statistic] \label{thm:representation population new} 
Assume Assumptions A.1-A.3, \ref{ass:CF1new}, and that $\theta $ satisfies Assumptions A.4 and \ref{ass:A5new}. Let $\rho \equiv rank(V[m(Z,\theta )])$.
\begin{enumerate}
\item If $\theta \in \partial \Theta _{I}$, then $Q_{n}(\theta )=H(\sqrt{n}\bar{Y}_{n})+\delta _{n}$, where
\begin{enumerate}
\item for any $C<\infty $, $P(|\delta _{n}|>Cn^{-1/2})=o(n^{-1/2})$.
\item $\bar{Y}_{n}:\Omega _{n}\to \mathbb{R}^{\rho }$ is a zero mean sample average of $n$ i.i.d.\ observations from a distribution with non-singular variance-covariance matrix $V=\mathbf{I}_{\rho }$ and finite fourth moments,
\item $H:\mathbb{R}^{\rho }\to \mathbb{R}$ is continuous, non-negative, weakly convex, and homogeneous of degree $\beta $. $H(y)=0$ implies for some non-zero vector $b\in \mathbb{R}^{\rho }$, $b^{\prime }y\leq 0$. For any $\mu >0$, any $|h|\geq \mu >0$, any $C_{2}>0$, any positive sequence $\{g_{n}\}_{n\geq 1}$, and any positive sequence $ \{\varepsilon _{n}\}_{n\geq 1}$ with $\varepsilon _{n}=o(1)$, $ \{H^{-1}(\{h\}^{\varepsilon _{n}})\cap \Vert y\Vert \leq C_{2}\sqrt{g_{n}} \}\subseteq \{H^{-1}(\{h\})\}^{\eta _{n}}$ where $\eta _{n}=O(\sqrt{g_{n}} \varepsilon _{n}),$ uniformly in $h$.
\item If we add Assumption \ref{ass:CF2new}, then for any $\mu >0$, any $ |h|\geq \mu >0$ and any sequence $\{\varepsilon _{n}\}_{n\geq 1}$ with $ \varepsilon _{n}=o(1)$, $\{H^{-1}(\{h\}^{\varepsilon _{n}})\}\subseteq \{H^{-1}(\{h\})\}^{\gamma _{n}}$ where $\gamma _{n}\leq O(\varepsilon _{n})$, uniformly in $h$.
\end{enumerate}
\item If $\theta \in Int(\Theta _{I})$, then $\lim \inf \{Q_{n}(\theta )=0\}$ a.s.
\end{enumerate}
\end{theorem}
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As a next step, we consider the standardized version of each of the approximating statistics in \cite{bugni:2014}: bootstrap, asymptotic approximation, subsampling 1, and subsampling 2. In each case, one can standardize resampling moment inequalities using: (a) the sample standard deviation or (b) the resampling standard deviation. Both of these options give the same formal results but the latter requires slightly longer arguments. For the sake of brevity, we express our approximation in terms of the first option. In particular:
\begin{itemize}
\item For the bootstrap, we replace \citet[Eq.~(3.2)]{bugni:2014} with
\begin{eqnarray*}
Q_{n}^{\ast }(\theta ) &=&G(\{[\sqrt{n}\left( \bar{m}_{n,j}^{\ast }(\theta )- \bar{m}_{n,j}(\theta )\right) /\hat{\sigma}_{n,j}(\theta )]_{-}\times 1[\bar{ m}_{n,j}(\theta )/\hat{\sigma}_{n,j}(\theta )\leq \tau _{n}/\sqrt{n} ]\}_{j=1}^{J}) \\
&=&G(\{[v_{n,j}^{\ast ,s}(\theta )]_{-}\times 1[\bar{m}_{n,j}(\theta )/\hat{ \sigma}_{n,j}(\theta )\leq \tau _{n}/\sqrt{n}]\}_{j=1}^{J}),
\end{eqnarray*}
where $\bar{m}_{n}^{\ast }(\theta )$ is the sample mean for the bootstrap sample.
\item For the asymptotic approximation, we replace \citet[Eq.~(4.2)]{bugni:2014} with
\begin{equation*}
Q_{n}^{AA}(\theta )=G\left( \left\{ \left[ \sqrt{n}\sum_{i=1}^{n}\zeta _{i}\left( m_{j}(Z_{i},\theta )-\bar{m}_{n,j}(\theta )\right) /\hat{\sigma} _{n,j}(\theta )\right] _{-}\times 1[ \bar{m}_{n,j}(\theta )/\hat{\sigma} _{n,j}(\theta )\leq \tau _{n}/\sqrt{n}] \right\} _{j=1}^{J}\right) ,
\end{equation*}
where $\{\zeta _{i}\}_{i=1}^{n}$ is an i.i.d.\ sample with $\zeta _{i}\sim N(0,1)$, independent of $\{Z_{i}\}_{i=1}^{n}$.
\item For subsampling 1, we replace \citet[Eq.~(5.2)]{bugni:2014} with
\begin{equation*}
Q_{b_{n},n}^{SS_{1}}(\theta )=G( \{ [ \sqrt{b_{n}}( \bar{ m}_{n,b_{n},j}^{SS}(\theta )-\bar{m}_{n,j}(\theta )) /\hat{\sigma} _{n,j}(\theta )] _{-}\times 1[ \bar{m}_{n,j}(\theta )/\hat{\sigma} _{n,j}(\theta )\leq \tau _{n}/\sqrt{n}]\} _{j=1}^{J}) ,
\end{equation*}
where $\bar{m}_{n,b_{n}}^{SS}(\theta )$ is the sample mean for the subsampling sample.
\item For subsampling 2, we replace \citet[Eq.~(5.4)]{bugni:2014} with
\begin{equation*}
Q_{b_{n},n}^{SS_{2}}(\theta )=G(\{[ \sqrt{b_{n}}\bar{m} _{n,b_{n},j}^{SS}(\theta )/\hat{\sigma}_{n,j}(\theta )] _{-}\} _{j=1}^{J}).
\end{equation*}
where $\bar{m}_{n,b_{n}}^{SS}(\theta )$ is the sample mean for the subsampling sample.
\end{itemize}

We now establish the analogous representation results for each of these standardized approximation methods. It is relevant to point out that the proofs of each of these build heavily on the corresponding representation results in \cite{bugni:2014}.
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\begin{theorem}[Representation result - Bootstrap] \label{thm:representation bootstrap new}
Assume Assumptions A.1-A.3, \ref{ass:CF1new}, and that $\theta $ satisfies Assumptions A.4-A.5. Let $\rho \equiv rank(V[m(Z,\theta )])$.
\begin{enumerate} \item If $\theta \in \partial \Theta _{I}$ then $Q_{n}^{\ast }(\theta )=H(\sqrt{n}\bar{Y}_{n}^{\ast })+\delta _{n}^{\ast }$, where
\begin{enumerate}
\item for any $C<\infty $, $P(|\delta _{n}^{\ast }|>Cn^{-1/2}|\mathcal{X} _{n})=o(n^{-1/2})$ a.s.
\item $\{\bar{Y}_{n}^{\ast }|\mathcal{X}_{n}\}:\Omega _{n}\to \mathbb{R}^{\rho }$ is a zero (conditional) mean sample average of $n$ i.i.d.\ observations from a distribution with a (conditional) variance-covariance matrix $\hat{V}$ which is non-singular a.s.\ and finite (conditional) fourth moments a.s., and $||\hat{V}-I_{p}||\leq O_{p}(n^{-1/2}) $.
\item $H:\mathbb{R}^{\rho }\to \mathbb{R}$ is the same function as in Theorem \ref{thm:representation population new}.
\end{enumerate}
\item If $\theta \in Int(\Theta _{I})$, then $\lim \inf \{Q_{n}^{\ast }(\theta )=0\}$ a.s.
\end{enumerate}
\end{theorem}
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\begin{theorem}[Representation result - AA] \label{thm:representationAA new} 
Assume Assumptions A.1-A.3 , \ref{ass:CF1new}, and that $\theta $ satisfies Assumptions A.4-A.5. Let $\rho \equiv rank(V[m(Z,\theta )])$.
\begin{enumerate}
\item If $\theta \in \partial \Theta _{I}$ then $Q_{n}^{AA}(\theta )=H(\sqrt{n}\bar{Y} _{n}^{AA})+\delta _{n}^{AA }$, where
\begin{enumerate}
\item for any $C<\infty $, $P(|\delta _{n}^{AA }|>Cn^{-1/2}|\mathcal{X} _{n})=o(n^{-1/2})$ a.s.
\item $\{\bar{Y}_{n}^{AA}|\mathcal{X}_{n}\}:\Omega _{n}\to \mathbb{R} ^{\rho }$ is a zero (conditional) mean sample average of $n$ i.i.d.\ observations from a distribution with a (conditional) variance-covariance matrix $\hat{V}$ which is non-singular a.s.\ and finite (conditional) fourth moments a.s., and $||\hat{V}-I_{p}||\leq O_{p}(n^{-1/2})$.
\item $H:\mathbb{R}^{\rho }\to \mathbb{R}$ is the same function as in Theorem \ref{thm:representation population new}.
\end{enumerate}
\item If $\theta \in Int(\Theta _{I})$, then $\lim \inf \{Q_{n}^{AA}(\theta )=0\}$ a.s.
\end{enumerate}
\end{theorem}
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\begin{theorem}[Representation result - SS1] \label{thm:representationSS1 new} 
Assume Assumptions A.1-A.3, \ref{ass:CF1new}, and that $\theta $ satisfies Assumptions A.4-A.5. Let $\rho \equiv rank(V[m(Z,\theta )])$.
\begin{enumerate}
\item If $\theta \in \partial \Theta _{I}$ then $Q_{b_{n},n}^{SS_{1}}(\theta )=H(\sqrt{b_{n}}\bar{Y} _{b_{n},n}^{SS})+\delta _{b_{n},n}^{SS_{1}}$, where
\begin{enumerate}
\item $\lim \inf \{P(\delta _{b_{n},n}^{SS_{1}}=0|\mathcal{X}_{n})\geq 1[||v_{n}(m_{\theta })||\leq \tau _{n}]\}$ and $\lim \inf \{\delta _{b_{n},n}^{SS_{1}}=0\},$ a.s.
\item $\{\bar{Y}_{b_{n},n}^{SS}|\mathcal{X}_{n}\}:\Omega _{n}\to \mathbb{R}^{\rho }$ is a zero (conditional) mean sample average of $b_{n}$ observations sampled without replacement from a distribution with a (conditional) variance-covariance matrix $\hat{V}$ which is non-singular a.s.\ and finite (conditional) fourth moments a.s., and $||\hat{V}-\mathbf{I} _{\rho }||\leq O_{p}(n^{-1/2})$.
\item $H:\mathbb{R}^{\rho }\to \mathbb{R}$ is the same function as in Theorem \ref{thm:representation population new}.
\end{enumerate}
\item If $\theta \in Int(\Theta _{I})$, then $\lim \inf \{Q_{b_{n},n}^{SS_{1}}(\theta )=0\}$ a.s.
\end{enumerate}
\end{theorem}
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\begin{theorem}[Representation result - SS2]\label{thm:representationSS2 new} 
Assume Assumptions A.1-A.3, \ref{ass:CF1new}, and that $\theta $ satisfies Assumptions A.4-A.5. Let $\rho \equiv rank(V[m(Z,\theta )])$.
\begin{enumerate}
\item If $\theta \in \partial \Theta _{I}$, then $Q_{b_{n},n}^{SS_{2}}(\theta )=H(\sqrt{b_{n}}\bar{Y} _{b_{n},n}^{SS})+\delta _{b_{n},n}^{SS_{2}}$, where
\begin{enumerate}
\item for some $C>0$, $P(|\delta _{b_{n},n}^{SS_{2}}|>C\sqrt{ (\ln \ln n)b_{n}/n}|\mathcal{X}_{n})=o(b_{n}^{-1/2})$ a.s.
\item $\{\bar{Y}_{b_{n},n}^{SS}|\mathcal{X}_{n}\}:\Omega _{n}\to \mathbb{R}^{\rho }$ is a zero (conditional) mean sample average of $b_{n}$ observations sampled without replacement from a distribution with a (conditional) variance-covariance matrix $\hat{V}$ which is non-singular a.s.\ and finite (conditional) fourth moments a.s., and $||\hat{V}-\mathbf{I} _{\rho }||\leq O_{p}(n^{-1/2})$.
\item $H:\mathbb{R}^{\rho }\to \mathbb{R}$ is the same function as in Theorem \ref{thm:representation population new}.
\end{enumerate}
\item If $\theta \in Int(\Theta _{I})$, then $\lim \inf \{Q_{b_{n},n}^{SS_{2}}(\theta )=0\}$ a.s.
\end{enumerate}
\end{theorem}
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\begin{remark}
	Notice that the representation result for the sample statistic (i.e.\ Theorem \ref{thm:representation population new}) is the only one of these results that requires slightly more than finite sixth absolute moments in Assumption \ref{ass:A5new}. All other representation results in this supplement can be established only using finite fourth absolute moments.
\end{remark}

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\section{Conclusion}

This supplement provides all the necessary results to show that a standardized version of the statistics considered by \cite{bugni:2014} satisfy the same conclusions regarding rate of convergence of the error in the coverage probability. Our strategy is to establish that all the representation results used in \cite{bugni:2014} for non-standardized statistics can also be established for standardized statistics, both for the sample statistic and for all approximating statistics (bootstrap, asymptotic approximation, subsampling 1, and subsampling 2). To establish these result we employ slightly longer formal arguments and we strengthen the finite moment requirements from finite fourth absolute moments to slightly more than finite sixth absolute moments. Using the representation results in this supplement, one can repeat the arguments in \cite{bugni:2014} to establish the exact same rates of convergence of the error in the coverage probability.

\newpage
\appendices

{\small 
\section{Appendix}
\subsection{Proofs of theorems}

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\begin{proof}[Proof of Theorem \ref{thm:representation population new}]

\underline{Part 1.} We show the result by slightly modifying the arguments in \citet[Part 1, Theorem A.1]{bugni:2014}. Let $\Sigma \equiv V[m(Z,\theta )]$ and, thus, $\rho =rank(V[m(Z,\theta )])$, and let $D\equiv Diag(\Sigma )$ and $\Omega \equiv D^{-1/2}\Sigma D^{-1/2}$. By definition, there are $(J-\rho )$ coordinates of $D^{-1/2}m(Z_{i},\theta )$ that can be expressed as a linear combination of the remaining $\rho $ coordinates of $ D^{-1/2}m(Z_{i},\theta )$ for all $i=1,\dots ,n$ (a.s.). We refer to these $\rho $ coordinates as the ``fundamental'' coordinates. Without loss of generality, we can rearrange $ D^{-1/2}m(Z_{i},\theta )$ s.t.\ the fundamental coordinates are the last $\rho$ ones. This implies that there is a matrix $ A\in \mathbb{R}^{(J-\rho )\times \rho }$ s.t.
\begin{equation}
D^{-1/2}m(Z_{i},\theta )=[A^{\prime },\mathbf{I}_{\rho }]^{\prime }\{m_{j}(Z_{i},\theta ) /\sigma_{j}(\theta) \}_{j=1}^{\rho }~\text{ for all }i=1,\dots,n,~ a.s.\label{eq:B-1b_sup}
\end{equation}
Let $\Omega_{\rho }$ denote the variance-covariance matrix of $ \{m_{j}(Z_{i},\theta ) /\sigma_{j}(\theta)\}_{j=1}^{\rho }$, which is necessarily positive definite and let $\Omega _{\rho }^{-1/2}$ denote the inverse of its square root. Then, we define $Y_{i}\equiv \Omega _{\rho }^{-1/2}\{(m_{j}(Z_{i},\theta )-E[m_{j}(Z_{i},\theta )])/\sigma_{j}(\theta)\}_{j=1}^{\rho }$ for all $i=1,\dots ,n$ and the matrix $B\equiv [A^{\prime },\mathbf{I}_{\rho }]^{\prime }\Omega _{\rho }^{1/2}\in \mathbb{R}^{J\times \rho }$. 

By these definitions and by Eq.\ \eqref{eq:B-1b_sup}, we then conclude that $BY_{i}=D^{-1/2}(m(Z_{i},\theta )-E[m(Z_{i},\theta )])$ for all $ i=1,\dots ,n$. According to this definition, $v_{n}^{s}(\theta )= \sqrt{n}B\bar{Y}_{n}$, $E[Y_{i}]=\mathbf{0}_{\rho }$, $V(Y_{i})=\mathbf{I}_{p}$, and $ E[|| Y_{i}|| ^{c}]<\infty $ for all $c>0$ s.t.\ $ E[|| m(Z,\theta )|| ^{c}]<\infty $. Moreover, $ \{m(Z_{i},\theta )\}_{i=1}^{n}$ are i.i.d., and so $\{Y_{i}\}_{i=1}^{n}$ are also i.i.d.

Let $B_{j}\in \mathbb{R}^{1\times \rho }$ denote the $j^{th}$ row of $B$. The function $H(y):\mathbb{R}^{\rho }\to \mathbb{R}$ is defined as $ H(y)\equiv G(\{[B_{j}y]_{-}1[E[m_{j}(Z,\theta )]=0]\}_{j=1}^{J})$. The same arguments in \citet[Theorem A.1]{bugni:2014} can be used to show that this function has all the desired properties. By definition,
\begin{equation*}
H(\sqrt{n}\bar{Y}_{n})\equiv G(\{[B_{j}\sqrt{n}\bar{Y}_{n}]_{-}1[E[m_{j}(Z, \theta )]=0]\}_{j=1}^{J}))=G(\{[v^{s}_{n,j}(\theta )]_{-}1[E[m_{j}(Z,\theta )]=0]\}_{j=1}^{J}))
\end{equation*}
and $\delta _{n}\equiv Q_{n}(\theta )-H(\sqrt{n}\bar{Y}_{n})$. In turn, $\delta _{n}=\delta _{n,1}+\delta _{n,2}$ where
\begin{eqnarray}
\delta _{n,1} &\equiv &Q_{n}(\theta )-G(\{\left[ \tilde{v}_{n,j}^{s}(\theta ) \right] _{-}1[E[m_{j}(Z,\theta )]=0]\}_{j=1}^{J}), \notag\\
\delta _{n,2} &\equiv &G(\{\left[ \tilde{v}_{n,j}^{s}(\theta )\right] _{-}1[E[m_{j}(Z,\theta )]=0]\}_{j=1}^{J})-G(\{\left[ v_{n,j}^{s}(\theta ) \right] _{-}1[E[m_{j}(Z,\theta )]=0]\}_{j=1}^{J}).\label{eq:deltan2}
\end{eqnarray}
We consider these two terms in separate steps. Step 1 shows that shows that $ P( |\delta _{n,1}|>0) =o(n^{-1/2})$ by a slightly modifying the arguments in \citet[Theorem A.1]{bugni:2014} and Step 2 shows that $P( |\delta _{n,2}|>Cn^{-1/2}) =o(n^{-1/2})$ for all $C<\infty $. The combination of these two steps completes the proof of this part.

\underline{Step 1: Argument for $\delta _{n,1}$.} Since $\theta \in \partial \Theta _{I}\subseteq \Theta _{I}$ then $ E[m(Z,\theta )]\geq \mathbf{0}_{J}$. Let $S\equiv \{j\in \{1,\dots ,J\}:E[m_{j}(Z,\theta )]=0\}$, $\bar{S}\equiv \{1,\dots ,J\}/S$ (which may be empty), $\eta \equiv \min_{j\in \bar{S}}E[m_{j}(Z,\theta )]/\sigma _{j}(\theta )>0$ if $\bar{S}\not=\emptyset $ and $\eta \equiv 0$ if $\bar{S} =\emptyset $. For any $j=1,\dots ,J$, consider the following argument. First, notice that:
\begin{eqnarray*}
[ \sqrt{n}\bar{m}_{n,j}(\theta )/\hat{\sigma}_{n,j}(\theta )]_{-} &=&[ \sqrt{n}\bar{m}_{n,j}(\theta )/\hat{\sigma}_{n,j}(\theta )]_{-}1[E[m_{j}(Z,\theta )]=0]+[\sqrt{n}\bar{m}_{n,j}(\theta )/\hat{\sigma} _{n,j}(\theta )]_{-}1[E[m_{j}(Z,\theta )]>0] \\ &\geq &[\tilde{v}_{n,j}(\theta )]_{-}1[E[m_{j}(Z,\theta )]=0].
\end{eqnarray*}
Second, notice that:
\begin{eqnarray*}
[\sqrt{n}\bar{m}_{n,j}(\theta )/\hat{\sigma}_{n,j}(\theta )]_{-}&=&[\sqrt{n} \bar{m}_{n,j}(\theta )/\hat{\sigma}_{n,j}(\theta )]_{-}1[E[m_{j}(Z,\theta )]=0]+[\sqrt{n}\bar{m}_{n,j}(\theta )/\hat{\sigma}_{n,j}(\theta )]_{-}1[E[m_{j}(Z,\theta )]>0] \\
&=&\left\{
\begin{array}{c}
	[ \tilde{v}_{n,j}^{s}(\theta )]_{-} 1[E[m_{j}(Z,\theta )]=0]  + \\ 
	 \lbrack v_{n,j}^{s}(\theta ) + \sqrt{n}E_{j}[m_{j}(Z,\theta )]/\sigma _{j}(\theta )\rbrack_{-}  1[E[m_{j}(Z,\theta )]>0](\sigma _{j}(\theta )/\hat{\sigma} _{n,j}(\theta ))
\end{array}
\right\} \\
&\leq& [\tilde{v}_{n,j}^{s}(\theta )]_{-}1[E[m_{j}(Z,\theta )]=0]+[v_{n,j}^{s}(\theta )+\sqrt{n}\eta ]_{-}1[E[m_{j}(Z,\theta )]>0](\sigma _{j}(\theta )/\hat{\sigma}_{n,j}(\theta )).
\end{eqnarray*}
From both of these, we extract the following conclusions. If $\bar{S} =\emptyset $ then $\delta _{n,1}=0$ and so $P(|\delta _{n,1}|>0)=0=o(n^{-1/2})$. If $\bar{S}\not=\emptyset $ then notice that $ \left\{ \min_{j\in \bar{S}}v_{n,j}^{s}(\theta )+\sqrt{n}\eta \geq 0\right\} \subseteq \{\delta _{n,1}=0\}$. Therefore, $P(|\delta _{n,1}|>0)\leq P(\min_{j\in \bar{S}}v_{n,j}^{s}(\theta )+\sqrt{n}\eta <0)$. Thus, the proof of this step is completed by showing that:
\begin{equation*}
P( \min_{j\in \bar{S}}v_{n,j}^{s}(\theta )+\sqrt{n}\eta <0) =o(n^{-1/2}).
\end{equation*}
To show this, notice that for any $\lambda \in (1/4,1/2)$ we have that:
\begin{eqnarray*}
P( \min_{j\in \bar{S}}v_{n,j}^{s}(\theta )+\sqrt{n}\eta <0) &=&P\left( \cup _{j\in \bar{S}}\{v_{n,j}^{s}(\theta )<-\sqrt{n}\eta \}\right) \leq \sum_{j\in \bar{S}}P\left( |v_{n,j}^{s}(\theta )|>\sqrt{n}\eta \}\right) \\
&\leq &\sum_{j\in \bar{S}}[1[n^{\lambda }>\sqrt{n}\eta ]+P(|v_{n,j}^{s}(\theta )|>n^{\lambda })]=o(n^{-1/2}),
\end{eqnarray*}
where the rate of convergence follows from $\eta >0$ ($S_{1}\not=\emptyset $) and Lemma \ref{lem:FirstSum_o}.

\underline{Step 2: Argument for $\delta _{n,2}$.} We show that $P\left( |\delta _{n,2}|>Cn^{-1/2}\right) =o(n^{-1/2})$ for all $C<\infty $. This part of the proof is the only genuinely new part of the argument relative to \citet[Theorem A.1]{bugni:2014}.

Fix $c>0$ arbitrarily small so that $\delta >(4+2\delta )2c+4$ and so $O(n^{1/2-\delta /4+\left( 1+\delta /2\right) 2c})=o(n^{-1/2})$. Then,
\begin{eqnarray*}
P(|\delta _{n,2}|>Cn^{-1/2}) 
&=&\left\{ 
\begin{array}{c}
P(\{|\delta _{n,2}|>Cn^{-1/2}\}\cap \{\cap _{j=1}^{J}\{\{|v_{n,j}^{s}(\theta )|\leq n^{c/2\beta }\}\cap \{|\hat{\sigma}_{n,j}(\theta )-\sigma _{j}(\theta )|\leq n^{-1/2-2c}\}\}\}) \\
+P(\{|\delta _{n,2}|>Cn^{-1/2}\}\cap \{\cup _{j=1}^{J}\{\{|v_{n,j}^{s}(\theta )|>n^{c/2\beta }\}\cup \{|\hat{\sigma} _{n,j}(\theta )-\sigma _{j}(\theta )|>n^{-1/2-2c}\}\}\})
\end{array}
\right\}  \\
&\leq& \sum_{j=1}^{J}P(|v_{n,j}^{s}(\theta )|>n^{c/2\beta })+\sum_{j=1}^{J}P(|\hat{\sigma}_{n,j}(\theta )-\sigma _{j}(\theta )|>n^{-1/2-2c}),
\end{eqnarray*}
\newline
where the inequality holds by Lemma \ref{lem:Preliminary2} for all sufficiently large $n$. The first sum is $o(n^{-1/2})$ by Lemma \ref{lem:FirstSum_o} (with $\lambda =c/2\beta $) and the second sum is $o(n^{-1/2})$ by Lemma \ref{lem:FinalBound} (with $\lambda =2c$). This completes the proof of this step.

\underline{Part 2.} If $\theta \in Int(\Theta _{I})$ then $E[m_{j}(Z,\theta )]>0$ for all $j=1,\dots ,J$. Define $\eta =\min_{j=1,\dots ,J}\{E[m_{j}(Z,\theta )]/\sigma _{j}(\theta )\}>0$. Consider any positive sequence $\{\varepsilon _{n}\}_{n\geq 1}$ s.t.\ $\sqrt{\ln \ln n}/\varepsilon _{n}=o(1)$ and $\varepsilon _{n}/\sqrt{n}=o(1)$. Suppose that the event $ \{\Vert v_{n}^{s}(m_{\theta })\Vert \leq \varepsilon _{n}\}$ occurs. Then, for all $n$ large enough, $Q_{n}(\theta )\leq G(\{[(-\varepsilon _{n}/\sqrt{n}+\eta )\sqrt{n}]_{-}\}_{j=1}^{J})=0$. Therefore, $\lim \inf \{\Vert v_{n}^{s}(m_{\theta })\Vert \leq \varepsilon _{n}\}\subseteq \lim \inf \{Q_{n}(\theta )=0\}$. By the LIL, $P(\lim \inf \{\Vert v_{n}^{s}(m_{\theta })\Vert \leq \varepsilon _{n}\})=1$ and the result follows.
\end{proof}
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\begin{proof}[Proof of Theorem \ref{thm:representation bootstrap new}]
\underline{Part 1.} We show the results by slightly modifying the arguments in \citet[Part 1, Theorem A.2]{bugni:2014}.
Let the matrices $\Sigma \in \mathbb{R}^{J\times J}$, $D\equiv Diag(\Sigma )$, $A\in \mathbb{R}^{(J-\rho )\times \rho }$, $\Omega _{\rho }\in \mathbb{R}^{\rho \times \rho }$, and $B\equiv [A^{\prime },\mathbf{I}_{\rho }]^{\prime }\Omega _{\rho }^{1/2}\in \mathbb{R}^{J\times \rho }$ be defined as in proof of Theorem \ref{thm:representation population new}. Also, define $\hat{\Sigma}\equiv \hat{V}[m(Z,\theta )]$ and $\hat{D }\equiv Diag(\hat{\Sigma})$.

Since bootstrap samples are constructed from the original random sample, it has to be the case that the coordinates of $\{m(Z_{i}^{\ast },\theta )\}_{i=1}^{n}$ can be arranged into the same $\rho $ ``fundamental'' and $(J-\rho )$ ``non-fundamental'' coordinates described in Theorem \ref{thm:representation population new}. In particular, the bootstrap sample satisfies Eq.\ \eqref{eq:B-1b_sup}. Then, by the argument in Theorem \ref{thm:representation population new}, we define $Y_{i}^{\ast }\equiv \Omega _{\rho }^{-1/2}\{(m_{j}(Z_{i}^{\ast },\theta )-\bar{m}_{n,j}(\theta ))/\hat{\sigma}_{j}(\theta)\}_{j=1}^{\rho }$ for all $i=1,\dots ,n$, which can be shown to satisfy $ BY_{i}^{\ast }=\hat{D}^{-1/2}(m(Z_{i}^{\ast },\theta )-\bar{m}_{n}(\theta ))$ for all $ i=1,\dots ,n$.

According to this definition of $\{Y_{i}\}_{i=1}^{n}$, $v_{n}^{s,\ast }(\theta )=\sqrt{n}B\bar{Y} _{n}^{\ast }$, $E[Y_{i}^{\ast }|\mathcal{X}_{n}]=\mathbf{0}_{\rho }$, $ V[Y_{i}^{\ast }|\mathcal{X}_{n}]=\hat{V}$ where $\hat{V} = \Omega_{\rho}^{-1/2} \hat{\Omega}_{\rho}\Omega_{\rho}^{-1/2}$, where $\hat{\Omega}_{\rho }$ denotes the sample correlation of $ \{ m_{j}(Z,\theta )\}_{j=1}^{\rho }$. By the SLLN, $\hat{\Sigma}-\Sigma =o(1)$ a.s.\ which implies that $ \hat{D}-D=o(1)$ a.s.\ and $\hat{\Omega}_{\rho }-\Omega_{\rho} =o(1)$ a.s. By this and the CMT, it then follows that $\hat{V}-\mathbf{I}_{\rho }=o(1)$ a.s., implying that $\hat{V}$ is non-singular, a.s.\ By a similar argument, the SLLN implies that $E(||Y_{i}^{\ast }||^{c}|\mathcal{X}_{n})<\infty $ for all $c>0$ s.t.\ $ E[\left\Vert m(Z,\theta )\right\Vert ^{c}]<\infty $. Moreover, $\{\{Y_{i}^{\ast }\}_{i=1}^{n}|\mathcal{X}_{n}\}$ are i.i.d.\ because $ \{\{m(Z_{i}^{\ast },\theta )\}_{i=1}^{n}|\mathcal{X}_{n}\}$ are i.i.d.  Finally, if we add Assumption A.5, the CLT and Slutzky's theorem imply that $\sqrt{n}(\hat{\Omega}-\hat{\Omega})=O_{p}(1)$ and, so, $||\hat{V} -\mathbf{I}_{\rho }||\leq O_{p}(n^{-1/2})$.

Let $H(y):\mathbb{R}^{\rho }\to \mathbb{R}$ be defined as in Theorem \ref{thm:representation population new}. The same arguments in \citet[Theorem A.1]{bugni:2014} can be used to show that this function has all the desired properties. By definition, it then follows that
\begin{equation*}
H(\sqrt{n}\bar{Y}_{n}^{\ast })\equiv G(\{[B_{j}\sqrt{n}\bar{Y}_{n}^{\ast }]_{-}1[E[m_{j}(Z,\theta )]=0]\}_{j=1}^{J}))=G(\{[v_{n,j}^{\ast ,s}(\theta )]_{-}1[E[m_{j}(Z,\theta )]=0]\}_{j=1}^{J}))
\end{equation*}
and $\delta _{n}^{\ast }\equiv Q_{n}^{\ast }(\theta )-H(\sqrt{n} \bar{Y}_{n}^{\ast })$. To conclude the proof, it suffices to show that this error term satisfies $P(|\delta _{n}^{\ast }|>0|\mathcal{X}_{n})=o(n^{-1/2})$. Since $\theta \in \partial \Theta _{I}\subseteq \Theta _{I}$ then $E[m_{j}(Z,\theta )]\geq 0$ $\forall j=1,\dots ,J$. Let $S\equiv \{j\in \{1,\dots ,J\}:E[m_{j}(Z,\theta )]>0\}$, $\bar{S}\equiv \{1,\dots ,J\}/S$ (which may be empty), $\eta \equiv \min_{j\in \bar{S} }E[m_{j}(Z,\theta )]/\sigma _{j}(\theta )>0$ if $\bar{S}\not=\emptyset $ and $\eta \equiv 0$ if $\bar{S}=\emptyset $, $\sigma _{L}^{2}(\theta )\equiv \min_{j=1,\dots ,J}\sigma _{j}^{2}(\theta )$, $\sigma _{H}^{2}(\theta )\equiv \max_{j=1,\dots ,J}\sigma _{j}^{2}(\theta )$, $C_{L}\equiv (1+\sigma _{L}^{2}(\theta )/(2\sigma _{H}^{2}))^{-1/2}$, and $C_{H}\equiv (1-\sigma _{L}^{2}(\theta )/(2\sigma _{H}^{2}))^{-1/2}>0$. Consider the following derivation:
\begin{eqnarray*}
&&\{\Vert v_{n}^{s}(m_{\theta })\Vert \leq \tau _{n}/C_{H}\}\cap \{||\hat{ \sigma}_{n}^{2}(\theta )-\sigma ^{2}(\theta )||\leq \sigma _{L}^{2}(\theta )/2\} \\
&&\subseteq \cap _{j=1}^{J}\{\{|\tilde{v}_{n,j}^{s}(m_{\theta })|\leq (\sigma _{j}(\theta )/\hat{\sigma}_{n,j}(\theta ))/(\tau _{n}/C_{H})\}\cap \{C_{L}\leq \sigma _{j}(\theta )/\hat{\sigma}_{n,j}(\theta )\leq C_{H}\}\} \\
&&\subseteq \cap _{j=1}^{J}\{\{\sqrt{n}|(\bar{m}_{n,j}(\theta )-E[m_{j}(Z,\theta )])|/\hat{\sigma}_{n,j}(\theta )\leq \tau _{n}\}\cap \{C_{L}\leq \sigma _{j}(\theta )/\hat{\sigma}_{n,j}(\theta )\leq C_{H}\}\} \\
&&\subseteq \{\cap _{j\in S}\{\sqrt{n}\bar{m}_{n,j}(\theta )/\hat{\sigma} _{n,j}(\theta )\leq \tau _{n}\}\}\cap \{\cap _{j\in \bar{S}}\{\sqrt{n}\bar{m} _{n,j}(\theta )/\hat{\sigma}_{n,j}(\theta )\geq C_{L}\sqrt{n}\eta -\tau _{n}\}\} \\
&&\subseteq \{\cap _{j\in S}\{\sqrt{n}\bar{m}_{n,j}(\theta )/\hat{\sigma} _{n,j}(\theta )\leq \tau _{n}\}\}\cap \{\cap _{j\in \bar{S}}\{\sqrt{n}\bar{m} _{n,j}(\theta )/\hat{\sigma}_{n,j}(\theta )>\tau _{n}\}\} =\{Q_{n}^{\ast }(\theta )=0\},
\end{eqnarray*}
where all inclusions are based on elementary arguments and the last inclusion holds for all sufficiently large $n$. From here, it then follows that
\begin{eqnarray*}
P(\delta _{n}^{\ast } =0|\mathcal{X}_{n})n^{1/2}&\leq& (P(\Vert v_{n}^{s}(m_{\theta })\Vert >\tau _{n}/C_{H}|\mathcal{X}_{n})+P(||\hat{\sigma }_{n}^{2}(\theta )-\sigma ^{2}(\theta )||>\sigma _{L}^{2}(\theta )/2| \mathcal{X}_{n})-1)n^{1/2} \\
&=&(1[\Vert v_{n}^{s}(m_{\theta })\Vert >\tau _{n}/C_{H}]+1[||\hat{\sigma} _{n}^{2}(\theta )-\sigma ^{2}(\theta )||>\sigma _{L}^{2}(\theta )/2])n^{1/2}.
\end{eqnarray*}
where the equality uses the fact that, conditionally on $\mathcal{X}_{n}$, the events $\{\Vert v_{n}^{s}(m_{\theta })\Vert >\tau _{n}/C_{H}\}$ and $\{|| \hat{\sigma}_{n}^{2}(\theta )-\sigma ^{2}(\theta )||>\sigma _{L}^{2}(\theta )/2\}$ are non-stochastic. To complete the proof, it suffices to show that $ P(\lim \inf \{\Vert v_{n}^{s}(m_{\theta })\Vert >\tau _{n}/C_{H}\})=1$ and $ P(\lim \inf \{||\hat{\sigma}_{n}^{2}(\theta )-\sigma ^{2}(\theta )||<\sigma _{L}^{2}(\theta )/2\})=1$. These two results follow from the LIL and the SLLN, respectively.

\underline{Part 2.} If $\theta \in Int(\Theta _{I})$ then $E[m_{j}(Z,\theta )]>0$ for all $j=1,\dots ,J$. Let $\eta \equiv \min_{j=1,\dots ,J}E[m_{j}(Z,\theta )]/\sigma _{j}(\theta )>0$, $\sigma _{L}^{2}(\theta )\equiv \min_{j=1,\dots ,J}\sigma _{j}^{2}(\theta )$, $\sigma _{H}^{2}(\theta )\equiv \max_{j=1,\dots ,J}\sigma _{j}^{2}(\theta )$, $ C_{L}\equiv (1+\sigma _{L}^{2}(\theta )/(2\sigma _{H}^{2}))^{-1/2}$, and $ C_{H}\equiv (1-\sigma _{L}^{2}(\theta )/(2\sigma _{H}^{2}))^{-1/2}>0$. By the same argument as in part 1, we have the following derivation:
\begin{eqnarray*}
&&\{\Vert v_{n}^{s}(m_{\theta })\Vert \leq \tau _{n}/C_{H}\}\cap \{||\hat{ \sigma}_{n}^{2}(\theta )-\sigma ^{2}(\theta )||\leq \sigma _{L}^{2}(\theta )/2\} \\
&& \subseteq \cap _{j=1}^{J}\{\{|\tilde{v}_{n,j}^{s}(m_{\theta })|\leq (\sigma _{j}(\theta )/\hat{\sigma}_{n,j}(\theta ))/(\tau _{n}/C_{H})\}\cap \{C_{L}\leq \sigma _{j}(\theta )/\hat{\sigma}_{n,j}(\theta )\leq C_{H}\}\} \\
&& \subseteq \cap _{j=1}^{J}\{\{\sqrt{n}|(\bar{m}_{n,j}(\theta )-E[m_{j}(Z,\theta )])|/\hat{\sigma}_{n,j}(\theta )\leq \tau _{n}\}\cap \{C_{L}\leq \sigma _{j}(\theta )/\hat{\sigma}_{n,j}(\theta )\leq C_{H}\}\} \\
&& \subseteq \cap _{j=1}^{J}\{\sqrt{n}\bar{m}_{n,j}(\theta )/\hat{\sigma} _{n,j}(\theta )\geq C_{L}\sqrt{n}\eta -\tau _{n}\} \\
&& \subseteq \cap _{j\in \bar{S}}\{\sqrt{n}\bar{m}_{n,j}(\theta )/\hat{\sigma} _{n,j}(\theta )>\tau _{n}\} = \{Q_{n}^{\ast }(\theta )=0\},
\end{eqnarray*}
where all inclusions are based on elementary arguments and the last inclusion holds for all sufficiently large $n$. From here, it then follows that
\begin{eqnarray*}
&&\{\lim \inf \{\Vert v_{n}^{s}(m_{\theta })\Vert \leq \tau _{n}/C_{H}\}\}\cap \{\lim \inf \{||\hat{\sigma}_{n}^{2}(\theta )-\sigma ^{2}(\theta )||\leq \sigma _{L}^{2}(\theta )/2\}\} \\
&&=\lim \inf \{\{\Vert v_{n}^{s}(m_{\theta })\Vert \leq \tau _{n}/C_{H}\}\cap \{||\hat{\sigma}_{n}^{2}(\theta )-\sigma ^{2}(\theta )||\leq \sigma _{L}^{2}(\theta )/2\}\}\subseteq \lim \inf \{Q_{n}^{\ast }(\theta )=0\}.
\end{eqnarray*}
To complete the proof, it suffices to show that $P(\lim \inf \{\Vert v_{n}^{s}(m_{\theta })\Vert >\tau _{n}/C_{H}\})=1$ and $P(\lim \inf \{||\hat{ \sigma}_{n}^{2}(\theta )-\sigma ^{2}(\theta )||<\sigma _{L}^{2}(\theta )/2\})=1$. These two results follow from the LIL and the SLLN, respectively.
\end{proof}
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\begin{proof}[Proof of Theorem \ref{thm:representationAA new}]
	This proof follows closely the arguments used to prove Theorem \ref{thm:representation bootstrap new}. The only difference is that we replace $\{Y_{i}\}_{i=1}^{n}$ with $\{Y_{i}^{AA }\}_{i=1}^{n}$ defined by $Y_{i}^{AA }\equiv \Omega _{\rho }^{-1/2}\{\zeta _{i}(m_{j}(Z_{i}^{\ast },\theta )-\bar{m}_{n,j}(\theta ))/\hat{\sigma}_{j}(\theta)\}_{j=1}^{\rho }$.
\end{proof}

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\begin{proof}[Proof of Theorem \ref{thm:representationSS1 new}]
	This proof follows closely the arguments used to prove Theorem \ref{thm:representation bootstrap new}. The only difference is that we replace $\{Y_{i}\}_{i=1}^{n}$ with $\{Y_{i}^{SS}\}_{i=1}^{b_{n}}$ defined by $Y_{i}^{SS }\equiv \Omega _{\rho }^{-1/2}\{(m_{j}(Z_{i}^{SS },\theta )-\bar{m}_{n,j}(\theta ))/\hat{\sigma}_{j}(\theta)\}_{j=1}^{\rho }$.
\end{proof}
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\begin{proof}[Proof of Theorem \ref{thm:representationSS2 new}]
We show the results by slightly modifying the arguments in \citet[Theorem A.15]{bugni:2014}. Since the structure of this proof is different from the one used to prove Theorems \ref{thm:representationAA new} or \ref{thm:representationSS1 new}, we cover the main differences.

\underline{Part 1.} Define $\delta _{b_{n},n}^{SS_{2}}\equiv Q_{b_{n},n}^{SS_{2}}(\theta )-G(\{[v_{b_{n},n}^{s,SS}(m_{j,\theta })]_{-}1[E[m_{j}(Z,\theta )]=0]\}_{j=1}^{J})$, where $ v_{b_{n},n}^{s,SS}(m_{j,\theta })\equiv \sqrt{b_{n}}(\bar{m} _{b_{n},n,j}^{SS}(\theta )-\bar{m}_{n,j}(\theta ))/\hat{\sigma}_{n,j}(\theta )$. Parts (b)-(c) follow from Theorem \ref{thm:representationSS1 new} so we focus on part (a).

For arbitrary $\delta \in (0,1/2)$, let $S\equiv \{j\in \{1,\dots ,J\}:E[m_{j}(Z,\theta )]=0\}$, $\bar{S}\equiv \{1,\dots ,J\}/S$, $\eta \equiv \min_{j\in \bar{S}}E[m_{j}(Z,\theta )]/\sigma _{j}(\theta )>0$, $ \sigma _{L}^{2}(\theta )\equiv \min_{j=1,\dots ,J}\sigma _{j}^{2}(\theta )$, $\lambda \equiv \max_{j=1,\dots ,J}(1+\delta )\sqrt{2}V[m_{j}(Z,\theta )]$, and let $A_{n}$ be defined as follows:
\begin{equation}
A_{n}\equiv \{|v_{b_{n},n}^{s,SS}(m_{j,\theta })|\leq b_{n}^{(1-\delta )/2}\}_{j=1}^{J}\cap \{|v_{n}^{s}(m_{j,\theta })|\leq \lambda \sqrt{\ln \ln n }\}_{j=1}^{J}\cap \{||\hat{\sigma}_{n}^{2}(\theta )-\sigma ^{2}(\theta )||\leq \sigma _{L}^{2}(\theta )/2\}\}. \label{eq:SS2-1_sup}
\end{equation}
For any $C>0$, notice that:
\begin{equation}
\sqrt{b_{n}}P(|\delta _{b_{n},n}^{SS_{2}}|>C\sqrt{(\ln \ln n)b_{n}/n}| \mathcal{X}_{n})\leq \sqrt{b_{n}}P(\{A_{n}\}^{c}|\mathcal{X}_{n})+\sqrt{b_{n} }P(\{|\delta _{b_{n},n}^{SS_{2}}|>C\sqrt{(\ln \ln n)b_{n}/n}\}\cap A_{n}| \mathcal{X}_{n}). \label{eq:SS2-2_sup}
\end{equation}
The proof is completed by showing that the two terms on the RHS of Eq.\ \eqref{eq:SS2-2_sup} are $o(1)$, a.s.

We begin with the first term in the RHS of Eq.\ \eqref{eq:SS2-2_sup}. Consider the following derivation:
\begin{eqnarray*}
\sqrt{b_{n}}P(\{A_{n}\}^{c}|\mathcal{X}_{n})  &=&\left\{ 
\begin{array}{c}
	\sum_{j=1}^{J}\sqrt{b_{n}} P(|v_{b_{n},n}^{s,SS}(m_{j,\theta })|>b_{n}^{(1-\delta )/2}|\mathcal{X} _{n})+\sum_{j=1}^{J}\sqrt{b_{n}}P(|v_{n}^{s}(m_{j,\theta })|>\lambda \sqrt{ \ln \ln n}|\mathcal{X}_{n})+\\
	\sum_{j=1}^{J}\sqrt{b_{n}}P(|\hat{\sigma} _{n,j}^{2}(\theta )-\sigma _{j}^{2}(\theta )|>\sigma _{L}^{2}(\theta )/2| \mathcal{X}_{n}) 
\end{array}
\right\} \\
&=& \left\{ 
\begin{array}{c}
	\sum_{j=1}^{J}\sqrt{b_{n}}P(|v_{b_{n},n}^{s,SS}(m_{j,\theta })|>b_{n}^{(1-\delta )/2}|\mathcal{X}_{n})+\sum_{j=1}^{J}\sqrt{b_{n}} 1(|v_{n}^{s}(m_{j,\theta })|>\lambda \sqrt{\ln \ln n})+\\
	\sum_{j=1}^{J}\sqrt{ b_{n}}1(|\hat{\sigma}_{n,j}^{2}(\theta )-\sigma _{j}^{2}(\theta )|>\sigma _{L}^{2}(\theta )/2),
\end{array}
\right\},
\end{eqnarray*}
where the first equality follows from elementary arguments and the second equality follows from the fact that $\{v_{n}(m_{j,\theta })|\mathcal{X} _{n}\} $ is deterministic. Fix $j=1,\dots ,J$ and $\varepsilon >0$ arbitrarily. By the LIL, $\lim \inf \{|v_{n}(m_{j,\theta })|\leq \lambda \sqrt{\ln \ln n}\}_{j=1}^{J}$ a.s.\ and so $P(\lim \sqrt{b_{n}} 1(|v_{n}(m_{j,\theta })|>\lambda \sqrt{\ln \ln n})=0)=1$. By the SLLN, $\lim \inf \{|\hat{\sigma}_{n,j}^{2}(\theta )-\sigma _{j}^{2}(\theta )|\leq \sigma _{L}^{2}(\theta )/2\}$ a.s.\ and so $P(\lim \sqrt{b_{n}}1(|\hat{\sigma} _{n,j}^{2}(\theta )-\sigma _{j}^{2}(\theta )|>\sigma _{L}^{2}(\theta )/2)=0)=1$. Next, consider the following derivation:
\begin{eqnarray*}
&&P(\lim \inf \{P(|v_{b_{n},n}^{s,SS}(m_{j,\theta })| >b_{n}^{(1-\delta )/2}| \mathcal{X}_{n})\sqrt{b_{n}}\leq \varepsilon \}) \\
&&\geq P\left( 
\begin{array}{c}
	\lim \inf \{P(|v_{b_{n},n}^{s,SS}(m_{j,\theta })|>b_{n}^{(1-\delta )/2}| \mathcal{X}_{n})b_{n}^{(\delta -1)}\leq (1+\varepsilon )\} \\
	\cap \lim \inf \{b_{n}^{(\delta -1/2)}(1+\varepsilon )\leq \varepsilon \}
\end{array}
\right) \\
&&\geq \left\{ 
\begin{array}{c}
	P(\lim \inf \{P(|v_{b_{n},n}^{s,SS}(m_{j,\theta })|>b_{n}^{(1-\delta )/2}| \mathcal{X}_{n})b_{n}^{(\delta -1)}\leq (1+\varepsilon )\}) \\
	+P(\lim \inf \{b_{n}^{(\delta -1/2)}(1+\varepsilon )\leq \varepsilon \})-1
\end{array}
\right\} \\
&&=P(\lim \inf \{P(|v_{b_{n},n}^{s,SS}(m_{j,\theta })|>b_{n}^{(1-\delta )/2}| \mathcal{X}_{n})b_{n}^{(\delta -1)}\leq (1+\varepsilon )\}) \leq P(\lim \inf \{1\leq 1+\varepsilon \})=1,
\end{eqnarray*}
where the two first two inequalities follow from elementary arguments, the following equality follows from $b_{n}^{(\delta -1/2)}=o(1)$, and the third inequality is shown later in Eq.\ \eqref{eq:SS2-3_sup}.

To complete this argument, consider the following derivation:
\begin{eqnarray*}
&&E(v_{b_{n},n}^{s,SS}(m_{j,\theta })^{2}|\mathcal{X}_{n}) \\
&&=\left( 
\begin{array}{c}
q_{n}^{-1}\sum_{s=1}^{q_{n}}b_{n}^{-1}\sum_{i=1}^{b_{n}}(m_{j}(X_{i,s}^{SS}, \theta )-\bar{m}_{n,j}(\theta ))^{2}/\hat{\sigma}_{n,j}^{2}(\theta )+ \\
q_{n}^{-1}\sum_{s=1}^{q_{n}}b_{n}^{-1}\sum_{a=1}^{b_{n}}\sum_{b=1,b \not=a}^{b_{n}}(m_{j}(X_{a,s}^{SS},\theta )-\bar{m}_{n,j}(\theta ))(m_{j}(X_{b,s}^{SS},\theta )-\bar{m}_{n,j}(\theta ))/\hat{\sigma} _{n,j}^{2}(\theta )
\end{array}
\right) \\
&&=\left( 
\begin{array}{c}
n^{-1}\sum_{i=1}^{n}(m_{j}(X_{i},\theta )-\bar{m}_{n,j}(\theta ))^{2}/\hat{ \sigma}_{n,j}^{2}(\theta )+ \\
+2n^{-1}(n-1)^{-1}\sum_{a=1}^{n}\sum_{b=1,b\not=a}^{n}(m_{j}(X_{a},\theta )- \bar{m}_{n,j}(\theta ))(m_{j}(X_{b},\theta )-\bar{m}_{n,j}(\theta ))/\hat{ \sigma}_{n,j}^{2}(\theta )
\end{array}
\right) \leq 1,
\end{eqnarray*}
where the first equality holds by expanding squares, the following equality holds by the fact that we are sampling without replacement, and the final inequality holds by verifying that, in the previous line, the first term equals one and the second term is non-positive by the negative associated produced by sampling without replacement (see \citet[Section 3.2(a)]{joag-dev/proschan:1983}). Chebyshev's inequality then implies that
\begin{equation}
1\geq E(v_{b_{n},n}^{s,SS}(m_{j,\theta })^{2}|\mathcal{X}_{n})\geq P(|v_{b_{n},n}^{s,SS}(m_{j,\theta })|>b_{n}^{(1-\delta )/2}|\mathcal{X} _{n})b_{n}^{(1-\delta )}, \label{eq:SS2-3_sup}
\end{equation}
which completes the proof for the first term on the RHS of Eq.\ \eqref{eq:SS2-2_sup}.

We now consider the second term RHS of Eq.\ \eqref{eq:SS2-2_sup}. Condition on $ \mathcal{X}_{n}$ and assume that $A_{n}$ occurs. Then,
\begin{eqnarray}
Q_{b_{n},n}^{SS_{2}}(\theta ) &=&G(\{[v_{b_{n},n}^{s,SS}(m_{j,\theta })+ \sqrt{b_{n}/n}v_{n}^{s}(m_{j,\theta })+\sqrt{b_{n}}E[m_{j}(Z,\theta )]/\hat{ \sigma}_{n,j}(\theta )]_{-}\}_{j=1}^{J}) \label{eq:SS2-4_sup} \\
&\leq &G(\{[v_{b_{n},n}^{s,SS}(m_{j,\theta })-\lambda \sqrt{(\ln \ln n)b_{n}/n}+\sqrt{b_{n}}E[m_{j}(Z,\theta )]/\hat{\sigma}_{n,j}(\theta )]_{-}\}_{j=1}^{J}) \notag \\
&\leq &G\left( \left\{ 
\begin{array}{c}
	{[ v_{b_{n},n}^{s,SS}(m_{j,\theta })-\lambda \sqrt{(\ln \ln n)b_{n}/n} ]}_{-}1[E[m_{j}(Z,\theta )]=0]+ \\
	{[ -b_{n}^{1/2-\delta }-\lambda \sqrt{(\ln \ln n)b_{n}/n}+\sqrt{b_{n}} 2\eta ]}_{-}1[E[m_{j}(Z,\theta )]>0]
\end{array}
\right\} _{j=1}^{J}\right)  \notag \\
&=&G(\{[v_{b_{n},n}^{s,SS}(m_{j,\theta })-\lambda \sqrt{(\ln \ln n)b_{n}/n} ]_{-}1[E[m_{j}(Z,\theta )]=0]\}_{j=1}^{J}) \notag \\
&\leq &G(\{[v_{b_{n},n}^{s,SS}(m_{j,\theta })]_{-}1[E[m_{j}(Z,\theta )]=0]\}_{j=1}^{J}+\lambda \sqrt{(\ln \ln n)b_{n}/n}1_{J}),
\end{eqnarray}
where the first equality is elementary, the second inequality holds because $ G$ is increasing and $A_{n}$ implies that $v_{n}^{s}(m_{j,\theta })\geq -\lambda \sqrt{\ln \ln n}$, the second inequality follows from the fact that $A_{n}$ implies that $v_{b_{n},n}^{s,SS}(m_{j,\theta })>-b_{n}^{(1-\delta )/2}$, the next equality follows for all $n$ large enough as $\eta ,\lambda ,\delta >0$, and the final inequality holds by elementary arguments. By using similar arguments, we can establish an analogous lower bound for $ Q_{b_{n},n}^{s,SS_{2}}(\theta )$. As a consequence, if $A_{n}$ occurs,
\begin{equation*}
Q_{b_{n},n}^{SS_{2}}(\theta )\in \left[ 
\begin{array}{c}
G(\{[v_{b_{n},n}^{s,SS}(m_{j,\theta })]_{-}1[E[m_{j}(Z,\theta )]=0]\}_{j=1}^{J}-\lambda \sqrt{(\ln \ln n)b_{n}/n}1_{J}), \\
G(\{[v_{b_{n},n}^{s,SS}(m_{j,\theta })]_{-}1[E[m_{j}(Z,\theta )]=0]\}_{j=1}^{J}+\lambda \sqrt{(\ln \ln n)b_{n}/n}1_{J})
\end{array}
\right]
\end{equation*}
By Assumption CF.2, $\forall x\in \mathbb{R}^{J}$ and $\forall \varepsilon >0$, $\exists D>0$ s.t.\ $|G(x+\varepsilon )-G(x)|\leq D||\varepsilon ||$. If we set $C=D\lambda >0$,
\begin{eqnarray*}
|\delta _{b_{n},n}^{SS_{2}}|&=&|Q_{b_{n},n}^{SS_{2}}(\theta )-G(\{[v_{b_{n},n}^{s,SS}(m_{j,\theta })]_{-}1[E[m_{j}(Z,\theta )]=0]\}_{j=1}^{J})| \\
&\leq &\max_{r\in \{-1,1\}}\left\{ \left\vert 
\begin{array}{c}
G(\{[v_{b_{n},n}^{s,SS}(m_{j,\theta })]_{-}1[E[m_{j}(Z,\theta )]=0]\}_{j=1}^{J}+r\lambda \sqrt{(\ln \ln n)b_{n}/n}1_{J}) \\
-G(\{[v_{b_{n},n}^{s,SS}(m_{j,\theta })]_{-}1[E[m_{j}(Z,\theta )]=0]\}_{j=1}^{J})
\end{array}
\right\vert \right\} \\
&\leq &D\Vert \lambda \sqrt{(\ln \ln n)b_{n}/n}\times 1_{J}\Vert =D\lambda \sqrt{(\ln \ln n)b_{n}/n}.
\end{eqnarray*}
This completes the argument for the second term on the RHS of Eq.\ \eqref{eq:SS2-2_sup}.

\underline{Part 2.} Since $Q_{b_{n},n}^{SS_{2}}(\theta )\geq 0$, it suffices to show that $\sqrt{b_{n}}P(Q_{b_{n},n}^{SS_{2}}(\theta )>0|\mathcal{X} _{n})=o(1)$. For arbitrary $\delta \in (0,1/2)$, let $\eta \equiv \min_{j=1,\dots ,J}E[m_{j}(Z,\theta )]/\sigma _{j}(\theta )>0$ and let $ A_{n}$ be defined as in Eq.\ \eqref{eq:SS2-1_sup}. By elementary arguments
\begin{equation*}
\sqrt{b_{n}}P(Q_{b_{n},n}^{SS_{2}}(\theta )>0|\mathcal{X}_{n})\leq \sqrt{ b_{n}}P(\{A_{n}\}^{c}|\mathcal{X}_{n})+\sqrt{b_{n}}P(\{Q_{b_{n},n}^{SS_{2}}( \theta )>0\}\cap A_{n}|\mathcal{X}_{n}).
\end{equation*}
We can now repeat arguments used in part 1 to argue that both terms on the RHS are $o(1)$, a.s.\ On the one hand, the same argument as in part 1 implies that $\sqrt{b_{n}}P(\{A_{n}\}^{c}|\mathcal{X}_{n})=o(1)$ a.s.\ On the other hand, if $A_{n}$ occurs, the argument used in Eq.\ \eqref{eq:SS2-4_sup} implies that $Q_{b_{n},n}^{SS_{2}}(\theta )\leq G(\{[-b_{n}^{1/2-\delta }-\lambda \sqrt{(\ln \ln n)b_{n}/n}+\eta 2\sqrt{b_{n}}]_{-}\}_{j=1}^{J})$. Since $\eta ,\delta >0$, the RHS expression is equal to zero for all $n$ large enough. This then implies that $\sqrt{b_{n}}P(\{Q_{b_{n},n}^{SS_{2}}( \theta )>0\}\cap A_{n}|\mathcal{X}_{n})=o(1)$, completing the proof.
\end{proof}
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\subsection{Proofs of intermediate results}
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\begin{lemma} \label{lem:Moments}
	Assume Assumption A.1 and that $\theta \in \Theta $ satisfies $E[||m(Z,\theta )||^{2+\psi }]<\infty $ for some $\psi >0$. For all $j=1,\dots ,J$,
\begin{eqnarray*}
E\left[ \left\vert m_{j}(Z_{i},\theta )-E[m_{j}(Z_{i},\theta )]\right\vert ^{2+\psi }\right] &<&\infty , \\
E\left[ \left\vert \left( m_{j}(Z_{i},\theta )-E[m_{j}(Z_{i},\theta )]\right) ^{2}-\sigma _{j}^{2}(\theta )\right\vert ^{1+\psi /2}\right] &<&\infty.
\end{eqnarray*}
\end{lemma}
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\begin{proof}
Fix $j=1,\dots ,J$ arbitrarily and let $\sigma ^{2}\equiv \sigma _{j}^{2}(\theta )$ and $M_{i}\equiv m_{j}(Z_{i},\theta )$ for all $ i=1,\dots ,n$.

By Assumption A.5 and Holder's inequality, $E[| M_{i}| ] ^{2+\psi }\leq E[ | M_{i}| ^{2+\psi }] <\infty $. Another application of Holder's inequality yields $|M_{i}-E[M_{i}]|^{2+\psi }\leq |M_{i}|^{2+\psi }+|E[M_{i}]^{2+\psi } $ and, so, $E|M_{i}-E[M_{i}]|^{2+\psi }\leq E|M_{i}|^{2+\psi }+|E[M_{i}]|^{2+\psi }<\infty $. This proves the first result.

Define $Y_{i}\equiv \left( M_{i}-E[M_{i}]\right) ^{2}/\sigma ^{2}-1$ and $ \phi \equiv \psi /2$. By definition,
\begin{equation*}
E[|(M_{i}-E[M_{i}])^{2}-\sigma ^{2}|^{1+\phi }]=\sigma ^{2+\phi }E\left[ \left\vert \frac{(M_{i}-E[M_{i}])^{2}}{\sigma ^{2}}-1\right\vert ^{1+\phi } \right] =\sigma ^{2+\phi }E[|Y_{i}|^{1+\phi }].
\end{equation*}
So, it suffices to show that $E[|Y_{i}|^{1+\phi }]<\infty $. For the remainder of the proof let $\beta >1$ be arbitrarily chosen. Notice that
\begin{eqnarray*}
E[|Y_{i}|^{1+\phi }] = E[ |Y_{i}|^{1+\phi }1[ |Y_{i}|>\beta]] +E[|Y_{i}|^{1+\phi }1[ |Y_{i}|\leq \beta ] ] \leq E[ |Y_{i}|^{1+\phi }1[ |Y_{i}|>\beta ]] +\beta ^{1+\phi }.
\end{eqnarray*}
So, it suffices to show that $E[ |Y_{i}| ^{1+\phi }1[ |Y_{i}| >\beta ] ] <\infty $. Since $\beta >1$, it follows that 
\begin{equation*}
\{|Y_{i}|>\beta \}\subseteq \left\{ \left\{ \left\vert \frac{ (M_{i}-E[M_{i}])^{2}}{\sigma ^{2}}\right\vert >\beta +1\right\} \cap \left\{ \left\vert Y_{i}\right\vert \leq \left\vert \frac{(M_{i}-E[M_{i}])^{2}}{ \sigma ^{2}}\right\vert \right\} \right\}
\end{equation*}
and, therefore,
\begin{eqnarray*}
E[|Y_{i}|^{1+\phi }1[|Y_{i}| >\beta ]]&\leq& E\left[ \left\vert \frac{ M_{i}-E[M_{i}]}{\sigma }\right\vert ^{2+\psi }1\left[ \frac{\left( M_{i}-E[M_{i}]\right) ^{2}}{\sigma ^{2}}>\beta +1\right] \right] \\
&\leq &E\left[ \left\vert \frac{M_{i}-E[M_{i}]}{\sigma }\right\vert ^{2+\psi }\right] =\frac{1}{\sigma ^{2+\psi }}E[|M_{i}-E[M_{i}]|^{2+\psi }]<\infty .
\end{eqnarray*}
This proves the second result and completes the proof.
\end{proof}
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\begin{remark} 
	Lemma \ref{lem:Moments} can be used with $\psi =2$ under Assumption A.5 or with $\psi>4$ under Assumption \ref{ass:A5new}. 
\end{remark}
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\begin{lemma} \label{lem:FirstSum_o} 
	Assume Assumption A.1 and that $\theta \in \Theta $ satisfies Assumption A.5. For all $j=1,\dots ,J$ and $c>0$, 
\begin{equation*}
	P[|v_{n,j}^{s}(\theta )|>n^{c}]=o(n^{-1/2}).
\end{equation*}
\end{lemma}
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\begin{proof}
For any $\psi >0$, consider the following argument based on Chebyshev's inequality:
\begin{eqnarray*}
P[|v_{n,j}^{s}(\theta )| >n^{c}]n^{(2+\psi )c}&=&P[|v_{n,j}^{s}(\theta )|^{2+\psi }>n^{(2+\psi )c}]n^{(2+\psi )c} \\
&\leq &E[|v_{n,j}^{s}(\theta )|^{2+\psi }]\\
&=& E\left[ \left\vert n^{-1/2}\sum_{i=1}^{n}\frac{[m_{j}(Z,\theta )-E[m_{j}(Z,\theta )]]}{\sigma _{j}(\theta )}\right\vert ^{2+\psi }\right] \\
&=&\sigma _{j}^{-(2+\psi )}(\theta )n^{-1-\psi /2}E\left[ \left\vert \sum_{i=1}^{n}[m_{j}(Z,\theta )-E[m_{j}(Z,\theta )]]\right\vert ^{2+\psi } \right] \\
&\leq &\sigma _{j}^{-(2+\psi )}(\theta )n^{-1-\psi /2}E\left[ \left( \sum_{i=1}^{n}|m_{j}(Z,\theta )-E[m_{j}(Z,\theta )]|^{2+\psi }\right) \right] \\
&\leq &\sigma _{j}^{-(2+\psi )}(\theta )n^{-\psi /2}E\left[ |m_{j}(Z,\theta )-E[m_{j}(Z,\theta )]|^{2+\psi }\right] .
\end{eqnarray*}
From this, Lemma \ref{lem:Moments} with $\psi =2$, and Assumption A.5, we conclude that
\begin{equation*}
P[|v_{n,j}^{s}(\theta )|>n^{c}]=O(n^{-\psi /2-(2+\psi )c})=o(n^{-1/2}),
\end{equation*}
where we have used that $c>0$ and $\psi \geq 1$.
\end{proof}
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\begin{lemma} \label{lem:Preliminary} 
	Assume Assumption \ref{ass:CF1new} and that $\theta \in \Theta _{I}$ satisfies Assumption A.4. Then, $\delta _{n,2}$ in Eq.\ \eqref{eq:deltan2} satisfies
\begin{equation*}
\left\vert \delta _{n,2}\right\vert \leq \left[ \left( 1-\tfrac{ \max_{j=1,\dots ,J}| \hat{\sigma}_{n,j}(\theta )-\sigma _{j}(\theta )| }{\min_{j=1,\dots ,J}\sigma _{j}(\theta )}\right) ^{-\beta }-1\right] G(\{ [ v_{n,j}(\theta )] _{-}1[E[m_{j}(Z,\theta )]=0]\} _{j=1}^{J}) ,
\end{equation*}
\end{lemma}
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\begin{proof}
For each $j=1,\dots ,J$, define $v_{n,j}^{L}(\theta )\equiv \min \{\tilde{v} _{n,j}^{s}(\theta ),v_{n,j}^{s}(\theta )\}$ and $v_{n,j}^{H}(\theta )\equiv \max \{\tilde{v}_{n,j}^{s}(\theta ),v_{n,j}^{s}(\theta )\}$. Since $\theta \in \Theta _{I}$, $E[m(Z,\theta )]\geq \mathbf{0}_{J}$. Define $ S_{0,-}(\theta )\equiv \{j=1,\dots ,J:E[m_{j}(Z,\theta )]=0\cap v_{n,j}^{L}(\theta )<0\}$ (which may be empty).

First, consider the case when $S_{0,-}(\theta )=\emptyset $. In this case, either $E[m_{j}(Z,\theta )]>0$ or $0\leq v_{n,j}^{L}(\theta )\leq v_{n,j}^{H}(\theta )$ for all $j=1,\dots ,J$. Therefore,
\begin{equation*}
\{ [\tilde{v}_{n,j}^{s}(\theta )]_{-}1[E[m_{j}(Z,\theta )]=0]\} _{j=1}^{J}=\{ [v_{n,j}^{s}(\theta )]_{-}1[E[m_{j}(Z,\theta )]=0]\} _{j=1}^{J}={\bf 0}_{J},
\end{equation*}
and, thus, $\delta _{n,2}=0$, and the statement holds.

Second, consider the case when $S_{0,-}(\theta )\not=\emptyset $. Define $ \alpha \equiv \max_{j\in S_{0,-}(\theta )}\{v_{n,j}^{L}(\theta )/v_{n,j}^{H}(\theta )\}\geq 1$. Now consider the following derivation:
\begin{eqnarray*}
	\alpha ^{\beta }G(\{ [v_{n,j}^{H}(\theta )]_{-}1[E[m_{j}(Z,\theta )]=0]\} _{j=1}^{J}) 
	&=&G( \{ [\alpha v_{n,j}^{H}(\theta )]_{-}1[E[m_{j}(Z,\theta )]=0]\} _{j\in S_{0,-}(\theta )},\{0\}_{j\not\in S_{0,-}(\theta )}) \\
	&\geq &G( \{ [v_{n,j}^{L}(\theta )]_{-}1[E[m_{j}(Z,\theta )]=0]\} _{j\in S_{0,-}(\theta )},\{0\}_{j\not\in S_{0,-}(\theta )}) \\
	&=&G( \{ [v_{n,j}^{L}(\theta )]_{-}1[E[m_{j}(Z,\theta )]=0]\} _{j=1}^{J})
\end{eqnarray*}
where the first equality follows from homogeneity of degree $\beta $ and the fact that $[\alpha v_{n,j}^{H}(\theta )]_{-}1[E[m_{j}(Z,\theta )]=0]=0$ for all $j\not\in S_{0,-}(\theta )$, the first inequality follows from the monotonicity of $G$ and $\alpha v_{n,j}^{H}(\theta )<\{v_{n,j}^{L}(\theta )/v_{n,j}^{H}(\theta )\}v_{n,j}^{H}(\theta )=v_{n,j}^{L}(\theta )$ for all $ j\in S_{0,-}(\theta )$, and the final equality follows from the fact that $ [v_{n,j}^{L}(\theta )]_{-}1[E[m_{j}(Z,\theta )]=0]=0$ for all $j\not\in S_{0,-}(\theta )$. From here, consider the following derivation:
\begin{eqnarray*}
	|\delta _{n,2}|&\leq& G( \{ [v_{n,j}^{L}(\theta )]_{-}1[E[m_{j}(Z,\theta )]=0]\} _{j=1}^{J}) -G( \{ [v_{n,j}^{H}(\theta )]_{-}1[E[m_{j}(Z,\theta )]=0]\} _{j=1}^{J}) \\
	&\leq &( \alpha ^{\beta }-1) G( \{ [v_{n,j}^{H}(\theta )]_{-}1[E[m_{j}(Z,\theta )]=0]\} _{j=1}^{J}) \\
&\leq &\left[ \left( \max_{j=1,\dots ,J}\left\{ \max \left\{ \frac{\hat{ \sigma}_{n,j}(\theta )}{\sigma _{j}(\theta )},\frac{\sigma _{j}(\theta )}{ \hat{\sigma}_{n,j}(\theta )}\right\} \right\} \right) ^{\beta }-1\right] G(\{ [v_{n,j}^{s}(\theta )]_{-}1[E[m_{j}(Z,\theta )]=0]\} _{j=1}^{J}) \\
&\leq &
\left\{
\begin{array}{c}
	\left[ \left( \max_{j=1,\dots ,J}\left\{ \max \left\{ \frac{ | \hat{\sigma}_{n,j}(\theta )-\sigma _{j}(\theta )| }{ \sigma _{L}(\theta )}+1,\frac{1}{1-\frac{| \hat{\sigma} _{n,j}(\theta )-\sigma _{j}(\theta )| }{\sigma _{L}(\theta )}} \right\} \right\} \right) ^{\beta }-1\right]  \\
\times 	G(\{ [v_{n,j}^{s}(\theta )]_{-}1[E[m_{j}(Z,\theta )]=0]\} _{j=1}^{J})
\end{array} 
\right\}\\
&\leq &\left[ \left( 1-\frac{\max_{j=1,\dots ,J}\left\vert \hat{\sigma} _{n,j}(\theta )-\sigma _{j}(\theta )\right\vert }{\sigma _{L}(\theta )} \right) ^{-\beta }-1\right] G(\{ [v_{n,j}^{s}(\theta )]_{-}1[E[m_{j}(Z,\theta )]=0]\} _{j=1}^{J}) ,
\end{eqnarray*}
where all inequalities are elementary and based on the definition of $ \alpha $ and the monotonicity of $G$.
\end{proof}

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\begin{lemma} \label{lem:Preliminary2} 
	Assume Assumption \ref{ass:CF1new} and that $\theta \in \Theta _{I}$ satisfies Assumption A.4. Then, $\forall c>0$, $ \exists N\in \mathbb{N}$ s.t.\ $\forall n\geq N$, $\delta _{n,2}$ in Eq.\ \eqref{eq:deltan2} satisfies
\begin{equation*}
\left\{ \left\{ \max_{j=1,\dots ,J}|\hat{\sigma}_{n,j}(\theta )-\sigma _{j}(\theta )|\leq n^{-1/2-2c}\right\} \cap \left\{ \max_{j=1,\dots ,k}|v_{n,j}^{s}(\theta )|\leq n^{c/2\beta }\right\} \right\} \subseteq \left\{ \left\vert \delta _{n,2}\right\vert \leq n^{-1/2-c}\right\} .
\end{equation*}
\end{lemma}
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\begin{proof} 
	First, assume that $\max_{j=1,\dots ,J}|\hat{\sigma}_{n,j}(\theta )-\sigma _{j}(\theta )|\leq n^{-1/2-2c}$. This implies that $\max_{j=1,\dots ,J}| \hat{\sigma}_{n,j}(\theta )-\sigma _{j}(\theta )|/\min_{j=1,\dots ,J}\sigma _{j}(\theta )\leq 1/2$ for all $n$ sufficiently large. For $x\in [0,1/2]$ consider the function $f(x)=(1-x)^{-\beta }-1$. By the intermediate value theorem, there is $\tilde{x}\in [0,1/2]$ s.t. $f(x)=f(0)+f^{\prime }(\tilde{x})x=\beta (1-\tilde{x})^{-(\beta +1)}x$ and, so, $| f(x)| <\beta 2^{(\beta +1)}x$. Therefore, 
\begin{equation*}
\left[ \left( 1-\frac{\max_{j=1,\dots ,J}|\hat{\sigma}_{n,j}(\theta )-\sigma _{j}(\theta )|}{\min_{j=1,\dots ,J}\sigma _{j}(\theta )}\right) ^{-\beta }-1\right] \leq \beta 2^{(\beta +1)}\max_{j=1,\dots ,J}|\hat{\sigma }_{n,j}(\theta )-\sigma _{j}(\theta )|\leq \beta 2^{(\beta +1)}n^{-1/2-2c}.
\end{equation*}

Second, assume that $\max_{j=1,\dots ,J}|v_{n,j}(\theta )| \leq n^{c/2\beta }$. If we combine this with the monotonicity and the homogeneity of $G$ we deduce that
\begin{equation*}
G(\{ [v_{n,j}^{s}(\theta )]_{-}1[E[m_{j}(Z,\theta )]=0]\} _{j=1}^{J}) \leq n^{c/2}G(\{1\}_{j=1}^{J}).
\end{equation*}

By combining these two steps with Lemma \ref{lem:Preliminary}, we conclude that $\max_{j=1,\dots ,J}|\hat{\sigma}_{n,j}(\theta )-\sigma _{j}(\theta )|\leq n^{-1/2-2c}$ and $\max_{j=1,\dots ,J}|v_{n,j}^{s}(\theta )|\leq n^{c/2\beta }$ imply that, for all $n$ sufficiently large,
\begin{equation*}
|\delta _{n,2}|\leq \beta 2^{(\beta +1)}n^{-1/2-2c}n^{c/2}G(\{1\}_{j=1}^{J})\leq \beta 2^{(\beta +1)}G(\{1\}_{j=1}^{J})n^{-1/2-1.5c}.
\end{equation*}
The RHS is less than $n^{-1/2-c}$ for all $n$ sufficiently large, and this completes the proof.
\end{proof}

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\begin{lemma} \label{lem:Bound1} 
	Assume Assumption A.1 and that $\theta \in \Theta $ satisfies $E[||m(Z,\theta )||^{2+\psi }]<\infty $ for some $\psi >0$. For all $j=1,\dots ,J$ and any sequence $\{a_{n}\}_{n\geq 1}=o(1)$,
\begin{equation*}
P( |\tilde{\sigma}_{n,j}(\theta )-\sigma _{j}(\theta )|>\sigma a_{n}) ( 3\sigma _{j}(\theta )a_{n}) ^{1+\psi /2}\leq n^{-\psi /2}E[| (m_{j}(Z_{i},\theta )-E[m_{j}(Z_{i},\theta )])^{2}-\sigma _{j}^{2}(\theta )|^{1+\psi /2}] .
\end{equation*}
\end{lemma}
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\begin{proof}
Consider the following argument: 
\begin{eqnarray*}
P\left( |\tilde{\sigma}_{n,j}(\theta )-\sigma _{j}(\theta )|>\sigma _{j}(\theta )a_{n}\right) &=&P\left( \left\{ \tilde{\sigma}_{n,j}(\theta )-\sigma _{j}(\theta )>\sigma _{j}(\theta )a_{n}\right\} \cup \left\{ \tilde{\sigma}_{n,j}(\theta )-\sigma _{j}(\theta )<-\sigma _{j}(\theta )a_{n}\right\} \right) \\
&=&P\left( \left\{ \tilde{\sigma}_{n,j}(\theta )>\sigma _{j}(\theta )\left( 1+a_{n}\right) \right\} \cup \left\{ \tilde{\sigma}_{n,j}(\theta )<\sigma _{j}(\theta )\left( 1-a_{n}\right) \right\} \right) \\
&= &
P\left(
\begin{array}{c}
	 \left\{ (\tilde{\sigma}_{n,j}^{2}(\theta )-\sigma _{j}^{2}(\theta ))>\sigma _{j}^{2}(\theta )(2a_{n}+a_{n}^{2})\right\} \cup  \\ 
	 \left\{ (\tilde{ \sigma}_{n,j}^{2}(\theta )-\sigma _{j}^{2}(\theta ))<\sigma _{j}^{2}(\theta )(-2a_{n}+a_{n}^{2})\right\} 
\end{array}
\right) \\
&\leq &
\left\{
\begin{array}{c}
	P\left( (\tilde{\sigma}_{n,j}^{2}(\theta )-\sigma _{j}^{2}(\theta ))>\sigma _{j}^{2}(\theta )( 2a_{n}+a_{n}^{2}) \right)\\ +P\left( ( \tilde{\sigma}_{n,j}^{2}(\theta )-\sigma _{j}^{2}(\theta ))<\sigma _{j}^{2}(\theta )( -2a_{n}+a_{n}^{2}) \right) 
\end{array}
\right\} \\
&\leq &P\left( (\tilde{\sigma}_{n,j}^{2}(\theta )-\sigma _{j}^{2}(\theta ))>3\sigma _{j}^{2}(\theta )a_{n}\right) +P\left( (\tilde{\sigma} _{n,j}^{2}(\theta )-\sigma _{j}^{2}(\theta ))>-3\sigma _{j}^{2}(\theta )a_{n}\right)\\
&\leq &P\left( |\tilde{\sigma}_{n,j}^{2}(\theta )-\sigma _{j}^{2}(\theta )|>3\sigma _{j}^{2}(\theta )a_{n}\right) ,
\end{eqnarray*}
where all relationships are elementary and we have used that $a_{n}=o(1)$ implies that there is $n$ large enough s.t.\ $2a_{n}+a_{n}^{2}<3a_{n}$ and $ -2a_{n}+a_{n}^{2}>-3a_{n}$. From here we conclude that
\begin{eqnarray*}
P( |\tilde{\sigma}_{n,j}(\theta )-\sigma _{j}(\theta )|>\sigma _{j}(\theta )a_{n}) ( 3\sigma _{j}(\theta )a_{n}) ^{1+\psi /2} &\leq &P( |\tilde{\sigma}_{n,j}^{2}(\theta )-\sigma _{j}^{2}(\theta )|>3\sigma _{j}^{2}(\theta )a_{n}) ( 3\sigma _{j}(\theta )a_{n}) ^{1+\psi /2} \\
&\leq &E( |\tilde{\sigma}_{n,j}^{2}(\theta )-\sigma _{j}^{2}(\theta )|^{1+\psi /2}) \\
&=&E\left( \left\vert \frac{\sum_{i=1}^{n}(m_{j}(Z_{i},\theta )-E[m_{j}(Z_{i},\theta )])^{2}-\sigma _{j}^{2}(\theta )}{n}\right\vert ^{1+\psi /2}\right) \\
&=&n^{-\psi /2}E\left( \frac{|\sum_{i=1}^{n}(m_{j}(Z_{i},\theta )-E[m_{j}(Z_{i},\theta )])^{2}-\sigma _{j}^{2}(\theta )|^{1+\psi /2}}{n} \right) \\
&\leq &n^{-\psi /2}E\left( \frac{\sum_{i=1}^{n}|(m_{j}(Z_{i},\theta )-E[m_{j}(Z_{i},\theta )])^{2}-\sigma _{j}^{2}(\theta )|^{1+\psi /2}}{n} \right) \\
&=&n^{-\psi /2}E\left[ |(m_{j}(Z_{i},\theta )-E[m_{j}(Z_{i},\theta )])^{2}-\sigma _{j}^{2}(\theta )|^{1+\psi /2}\right] ,
\end{eqnarray*}
where we have used Markov's and Holder's inequalities.
\end{proof}
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\begin{lemma} \label{lem:Bound2} 
	Assume Assumption A.1 and that $\theta \in \Theta $ satisfies $E[||m(Z,\theta )||^{2+\psi }]<\infty $ for some $\psi >0$. For all $j=1,\dots ,J$ and any sequence $\{a_{n}\}_{n\geq 1}=o(1)$,
\begin{eqnarray*}
&&P( |\tilde{\sigma}_{n,j}(\theta )-\hat{\sigma}_{n,j}(\theta )|>a_{n}) (3\sigma _{j}(\theta )a_{n})^{1+\psi /2} \\
&&\leq n^{-1-\psi }E[|m_{j}(Z_{i},\theta )-E[m_{j}(Z_{i},\theta )]|^{2+\psi }]+n^{-\psi /2}E[|(m_{j}(Z_{i},\theta )-E[m_{j}(Z_{i},\theta )])^{2}-\sigma _{j}^{2}(\theta )|^{1+\psi /2}].
\end{eqnarray*}
\end{lemma}
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\begin{proof}
First, consider the following argument: 
\begin{eqnarray*}
&&P\left( |\tilde{\sigma}_{n,j}(\theta )-\hat{\sigma}_{n,j}(\theta )|>a_{n}\right) (3\sigma _{j}(\theta )a_{n})^{1+\psi /2}\\
 &&=P(|\tilde{\sigma}_{n,j}^{2}(\theta )-\hat{\sigma}_{n,j}^{2}(\theta )|^{1+\psi /2}>(3\sigma _{j}(\theta )a_{n})^{1+\psi /2})(3\sigma _{j}(\theta )a_{n})^{1+\psi /2} \\
&&\leq E(|\tilde{\sigma}_{n,j}^{2}(\theta )-\hat{\sigma}_{n,j}^{2}(\theta )|^{1+\psi /2}) \\
&&=E\left( \left\vert \frac{\sum_{i=1}^{n}[(m_{j}(Z_{i},\theta )-\bar{m} _{n,j}(\theta ))^{2}-(m_{j}(Z_{i},\theta )-E[m_{j}(Z_{i},\theta )])^{2}]}{n} \right\vert ^{1+\psi /2}\right) \\
&&=E(|\bar{m}_{n,j}(\theta )-E[m_{j}(Z_{i},\theta )]|^{2+\psi }) \leq n^{-1-\psi }E(|m_{j}(Z_{i},\theta )-E[m_{j}(Z_{i},\theta )]|^{2+\psi }),
\end{eqnarray*}
where every relationship is elementary. Now, consider the following derivation:
\begin{eqnarray*}
&&P(|\tilde{\sigma}_{n,j}(\theta )-\hat{\sigma}_{n,j}(\theta )| >a_{n}) \\
&&=P(\{\hat{\sigma}_{n,j}(\theta )>a_{n}+\tilde{\sigma}_{n,j}(\theta )\}\cup \{\hat{\sigma}_{n,j}(\theta )<-a_{n}+\tilde{\sigma}_{n,j}(\theta )\}) \\
&&=P(\{\hat{\sigma}_{n,j}^{2}(\theta )-\tilde{\sigma}_{n,j}^{2}(\theta )>a_{n}^{2}+2a_{n}\tilde{\sigma}_{n,j}(\theta )\}\cup \{\hat{\sigma} _{n,j}^{2}(\theta )-\tilde{\sigma}_{n,j}^{2}(\theta )<a_{n}^{2}-2a_{n}\tilde{ \sigma}_{n,j}(\theta )\}) \\
% &&=\left\{ 
%\begin{array}{c}
%P(\{\hat{\sigma}_{n,j}^{2}(\theta )-\tilde{\sigma}_{n,j}^{2}(\theta )>a_{n}^{2}+2a_{n}\tilde{\sigma}_{n,j}(\theta )\}\cup \{\hat{\sigma} _{n,j}^{2}(\theta )-\tilde{\sigma}_{n,j}^{2}(\theta )<a_{n}^{2}-2a_{n}\tilde{ \sigma}_{n,j}(\theta )\}\cap \{|\tilde{\sigma}_{n,j}(\theta )-\hat{\sigma} _{n,j}(\theta )|\leq \sigma _{j}(\theta )a_{n}\}) \\
%+P(\{\hat{\sigma}_{n,j}^{2}(\theta )-\tilde{\sigma}_{n,j}^{2}(\theta )>a_{n}^{2}+2a_{n}\tilde{\sigma}_{n,j}(\theta )\}\cup \{\hat{\sigma} _{n,j}^{2}(\theta )-\tilde{\sigma}_{n,j}^{2}(\theta )<a_{n}^{2}-2a_{n}\tilde{ \sigma}_{n,j}(\theta )\}\cap \{|\tilde{\sigma}_{n,j}(\theta )-\hat{\sigma} _{n,j}(\theta )|>\sigma _{j}(\theta )a_{n}\})
%\end{array}
%\right\}  \\
&&\leq \left\{ 
\begin{array}{c}
P(\{\hat{\sigma}_{n,j}^{2}(\theta )-\tilde{\sigma}_{n,j}^{2}(\theta )>a_{n}^{2}(1+2\sigma _{j}(\theta ))+2a_{n}\sigma _{j}(\theta )\}\cup \{\hat{ \sigma}_{n,j}^{2}(\theta )-\tilde{\sigma}_{n,j}^{2}(\theta )<a_{n}^{2}a_{n}^{2}(1+2\sigma _{j}(\theta ))-2a_{n}\sigma _{j}(\theta )\}) \\
+P(|\tilde{\sigma}_{n,j}(\theta )-\hat{\sigma}_{n,j}(\theta )|>\sigma _{j}(\theta )a_{n})
\end{array}
\right\}  \\
&& \leq P(|\hat{\sigma}_{n,j}^{2}(\theta )-\tilde{\sigma}_{n,j}^{2}(\theta )|>3a_{n}^{2}\sigma _{j}(\theta ))+P(|\tilde{\sigma}_{n,j}(\theta )-\hat{ \sigma}_{n,j}(\theta )|>\sigma _{j}(\theta )a_{n}),
\end{eqnarray*}
where all relationships are elementary and we have used that $ a_{n}=o(1)$ implies that there is $n$ large enough s.t.\ $ a_{n}^{2}(1+2\sigma _{j}(\theta ))+2a_{n}\sigma _{j}(\theta )<3a_{n}\sigma _{j}(\theta )$ and $a_{n}^{2}(1+2\sigma _{j}(\theta ))-2a_{n}\sigma _{j}(\theta )>-3a_{n}\sigma _{j}(\theta )$. From here we conclude that
\begin{eqnarray*}
&&P(|\tilde{\sigma}_{n,j}(\theta )-\hat{\sigma}_{n,j}(\theta )|>a_{n})(3\sigma _{j}(\theta )a_{n})^{1+\psi /2} \\
&&\leq P(|\hat{\sigma}_{n,j}^{2}(\theta )-\hat{\sigma}_{n,j}^{2}(\theta )|>3a_{n}\sigma _{j}(\theta ))(3\sigma _{j}(\theta )a_{n})^{1+\psi /2}+P(| \tilde{\sigma}_{n,j}(\theta )-\sigma _{j}(\theta )|>\sigma _{j}(\theta )a_{n})(3\sigma _{j}(\theta )a_{n})^{1+\psi /2} \\
&& \leq n^{-1-\psi }E(|m_{j}(Z_{i},\theta )-E[m_{j}(Z_{i},\theta )]|^{2+\psi })+n^{-\psi /2}E[|(m_{j}(Z_{i},\theta )-E[m_{j}(Z_{i},\theta )])^{2}-\sigma _{j}^{2}(\theta )|^{1+\psi /2}],
\end{eqnarray*}
where we have used Lemma \ref{lem:Bound1}.
\end{proof}
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\begin{lemma} \label{lem:Bound3} 
	Assume Assumption A.1 and that $\theta \in \Theta $ satisfies $E[||m(Z,\theta )||^{2+\psi }]<\infty $ for some $\psi >0$. For all $j=1,\dots ,J$ and any sequence $\{a_{n}\}_{n\geq 1}=o(1)$,
\begin{eqnarray*}
&&P(|\hat{\sigma}_{n,j}(\theta )-\sigma _{j}(\theta )|>2a_{n})(3\sigma _{j}(\theta )a_{n})^{1+\psi /2} \\
&&\leq n^{-1-\psi }E[|m_{j}(Z_{i},\theta )-E[m_{j}(Z_{i},\theta )]|^{2+\psi }]+n^{-\psi /2}2E[|(m_{j}(Z_{i},\theta )-E[m_{j}(Z_{i},\theta )])^{2}-\sigma _{j}^{2}(\theta )|^{1+\psi /2}].
\end{eqnarray*}
\end{lemma}
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\begin{proof}
By triangular inequality, $|\hat{\sigma}_{n,j}(\theta )-\sigma _{j}(\theta )|\leq |\hat{\sigma}_{n,j}(\theta )-\tilde{\sigma}_{n,j}(\theta )|+|\tilde{ \sigma}_{n,j}(\theta )-\sigma _{j}(\theta )|$ and therefore
\begin{eqnarray*}
P(|\hat{\sigma}_{n,j}(\theta )-\sigma _{j}(\theta )| >2a_{n}) &\leq& P(\{| \hat{\sigma}_{n,j}(\theta )-\tilde{\sigma}_{n,j}(\theta )|>a_{n}\}\cup \{| \tilde{\sigma}_{n,j}(\theta )-\sigma _{j}(\theta )|>a_{n}\}) \\
&\leq &P(|\hat{\sigma}_{n,j}(\theta )-\tilde{\sigma}_{n,j}(\theta )|>a_{n})+P(|\tilde{\sigma}_{n,j}(\theta )-\sigma _{j}(\theta )|>a_{n}).
\end{eqnarray*}
The result follows from the previous inequality and Lemmas \ref{lem:Bound1} and \ref{lem:Bound2} .
\end{proof}
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\begin{lemma}\label{lem:FinalBound}
	Assume Assumption A.1 and that $\theta \in \Theta $ satisfies $E[||m(Z,\theta )||^{2+\psi }]<\infty $ for some $\psi >0$. For all $ j=1,\dots ,J$ and any $c>0$,
\begin{equation*}
P(|\hat{\sigma}_{n,j}(\theta )-\sigma _{j}(\theta )>n^{-1/2-c})=O(n^{1/2-\psi /4+\left( 1+\psi /2\right) c}).
\end{equation*}
Under Assumption \ref{ass:A5new} the RHS expression is $o(n^{-1/2})$ for a choice of $c>0$ that is small enough.
\end{lemma}
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\begin{proof}
Let $c \in (0,1/2)$. By Lemmas \ref{lem:Moments} and \ref{lem:Bound3} applied to $a_{n}=n^{-1/2-c}/2=o(1)$, we conclude that
\begin{equation*}
P(|\hat{\sigma}_{n,j}(\theta )-\sigma _{j}(\theta )|>n^{-1/2-c})=(3\sigma n^{-1/2-c}/2)^{-1-\psi /2}O(n^{-1-\psi }+n^{-\psi /2})=O(n^{1/2-\psi /4+( 1+\psi /2) c}).
\end{equation*}
To complete the proof, we notice that $O( n^{1/2-\psi /4+( 1+\psi /2) c}) =o(n^{-1/2})$ if and only if $\psi >( 4+2\psi) c+4$. By making $c$ arbitrarily small, the condition can be achieved whenever $\psi >4$, i.e., under Assumption \ref{ass:A5new}.
\end{proof}
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