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\begin{document}

\title{\textbf{Online supplementary material to ``Structural change in non-stationary AR(1) models"}\textbf{\thanks{Tianxiao Pang and Yanling Liang's research was supported by the Department
of Education of Zhejiang Province in China (N20140202). \newline
\hspace*{5mm}E-mail addresses: txpang@zju.edu.cn, chong2064@cuhk.edu.hk,
zhangdanna0507@gmail.com and 1040857986@qq.com.}}\\
}
\date{\today }
\author{}
\maketitle

\begin{center}
\vskip-1cm {{\small Tianxiao Pang$^{1,}$\footnote[2]{Corresponding author.}, Terence Tai-Leung Chong$^{2,3}$, Danna Zhang$^{4}$
and Yanling Liang$^{1}$, }}
{\small \centerline{$^{1}$School of Mathematical Sciences, Yuquan Campus, Zhejiang University, Hangzhou 310027, P.R.China}}
{\small \centerline{$^{2}$Department of Economics and Lau Chor Tak Institute of Global Economics and Finance, }}
{\small \centerline{The Chinese University of Hong Kong, Hong Kong}}
{\small \centerline{$^{3}$Department of International Economics and Trade, Nanjing University, Nanjing 210093, P.R.China}}
{\small \centerline{$^{4}$Department of Statistics, University of Chicago, 5734 S. University Ave., Chicago, Illinois 60637, USA}}
\end{center}
\bigskip


\renewcommand{\thesection}{S} \renewcommand\theequation{S.\arabic{equation}} \setcounter{equation}{0}

\noindent \textbf{Proof of Lemma A.1}. \ The proofs of (a) and (b) can be completed by following the proof of Theorem 3.2 in Phillips and Magdalinos (2007a) and by using the truncation technique as shown in (A.1). The details are omitted. To prove part (c), it suffices to note that
\begin{eqnarray*}
\frac{y_{[rT]}}{\sqrt{k_{T}l(\eta _{T})}} &=&\frac{1}{\sqrt{k_{T}l(\eta_{T})}}\Big((1-\frac{c}{k_{T}})^{[rT]}y_{0}+\sum_{i=0}^{[rT]-1}(1-\frac{c}{k_{T}})^{i}\varepsilon_{\lbrack rT]-i}\Big) \\
&=&\sum_{i=0}^{[rT]-1}\big((1-\frac{c}{k_{T}})^{k_{T}}\big)^{i/k_{T}}\frac{\varepsilon_{\lbrack rT]-i}}{\sqrt{k_{T}l(\eta _{T})}}+o_{p}(1) \\
&\Rightarrow &\int_{0}^{\infty}\exp {(-cs)}dW(s)
\end{eqnarray*}%
by Theorem 1 in Cs\"{o}rg\H{o} et al. (2003), (3.4) in Gin\'{e} et al. (1997) and assumptions C2 and C3. $\hfill \Box $\newline


\noindent \textbf{Proof of Lemma A.2}. \ To prove part (a), we first note that
\begin{equation*}
y_{t-1}\varepsilon _{t}=\frac{1}{2}(y_{t}^{2}-y_{t-1}^{2}-\varepsilon_{t}^{2}),~~~t=[\tau _{0}T]+1,\cdots ,T,
\end{equation*}
which implies that
\begin{equation}\label{1.1}
\frac{1}{Tl(\eta_{T})}\sum_{t=[\tau_{0}T]+1}^{T}y_{t-1}\varepsilon_{t}=\frac{1}{2Tl(\eta_{T})}\Big(y_{T}^{2}-y_{[\tau_{0}T]}^{2}-\sum_{t=[\tau_{0}T]+1}^{T}\varepsilon _{t}^{2}\Big).
\end{equation}
Note also that
\begin{equation}\label{1.3}
\frac{y_{[\tau_{0}T]}}{\sqrt{Tl(\eta _{T})}}=o_{p}(1)
\end{equation}
by part (c) of Lemma A.1. We let
\begin{equation*}
S_{T}(\tau _{0},s):=\frac{1}{\sqrt{Tl(\eta _{T})}}\sum_{t=[\tau_{0}T]+1}^{[sT]}\varepsilon _{t}.
\end{equation*}
Then
\begin{equation}  \label{1.2}
S_{T}(\tau _{0},s)\Rightarrow \overline{W}(s)-\overline{W}(\tau _{0})
\end{equation}
for any $\tau _{0}\leq s\leq 1$ by the functional central limit theorem for the i.i.d. random variables from the DAN (see Theorem 1 in Cs\"{o}rg\H{o} et
al. (2003) and (3.4) in Gin\'{e} et al. (1997)). Combining this result with (\ref{1.3}) immediately leads to
\begin{equation}  \label{1.12}
\frac{y_{T}^{2}-y_{[\tau _{0}T]}^{2}}{2Tl(\eta _{T})}=\frac{(y_{[\tau_{0}T]}+\sum_{i=[\tau _{0}T]+1}^{T}\varepsilon _{i})^{2}-y_{[\tau _{0}T]}^{2}}{2Tl(\eta _{T})}\Rightarrow \frac{1}{2}(\overline{W}(1)-\overline{W}(\tau_{0}))^{2}.
\end{equation}
In addition, it is trivial that
\begin{equation}  \label{1.11}
\frac{\sum_{t=[\tau_{0}T]+1}^{T}\varepsilon _{t}^{2}}{2Tl(\eta_{T})}\overset{p}{\rightarrow} \frac{1-\tau_{0}}{2}
\end{equation}
when $\varepsilon_{t}^{\prime }s$ are i.i.d. random variables in the DAN. Now, it follows from (\ref{1.1}), (\ref{1.12}) and (\ref{1.11}) that part (a) holds.

To prove part (b), using (\ref{1.3}) and (\ref{1.2}), we have
\begin{eqnarray}  \label{1.9}
&&\frac{1}{T^{2}l(\eta _{T})}\sum_{t=[\tau _{0}T]+1}^{T}y_{t-1}^{2}  \notag\\
&=&\frac{1}{T^{2}l(\eta _{T})}\sum_{t=[\tau _{0}T]+1}^{T}\Big(y_{[\tau_{0}T]}+\sum_{i=[\tau _{0}T]+1}^{t-1}\varepsilon _{i}\Big)^{2}  \notag \\
&=&\frac{1}{T}\sum_{t=[\tau _{0}T]+1}^{T}\Big(\frac{y_{[\tau _{0}T]}}{\sqrt{Tl(\eta _{T})}}+S_{T}(\tau _{0},\frac{t-1}{T})\Big)^{2}  \notag \\
&=&\sum_{t=[\tau _{0}T]+1}^{T}\int_{\frac{t-1}{T}}^{\frac{t}{T}}\Big(\frac{y_{[\tau _{0}T]}}{\sqrt{Tl(\eta _{T})}}+S_{T}(\tau _{0},\frac{t-1}{T})\Big)^{2}ds  \notag \\
&=&\sum_{t=[\tau _{0}T]+1}^{T}\int_{\frac{t-1}{T}}^{\frac{t}{T}}\Big(\frac{y_{[\tau _{0}T]}}{\sqrt{Tl(\eta _{T})}}+S_{T}(\tau _{0},s)\Big)^{2}ds\cdot
(1+o_{p}(1))  \notag \\
&=&\int_{\tau _{0}}^{1}\Big(\frac{y_{[\tau _{0}T]}}{\sqrt{Tl(\eta _{T})}}+S_{T}(\tau _{0},s)\Big)^{2}ds\cdot (1+o_{p}(1))  \notag \\
&\Rightarrow &\int_{\tau _{0}}^{1}(\overline{W}(s)-\overline{W}(\tau_{0}))^{2}ds.
\end{eqnarray}

Finally, since part (a) and part (b) hold jointly, this completes our proof. $\hfill \Box $\newline



\noindent \textbf{Proof of Lemma A.3}. \ Noting that the fact $|\hat{\tau}_{T}-\tau _{0}|=O_{p}(k_{T}/T)$ is proved in the proof of part (a) of Theorem 1.1 and the limiting distribution in Lemma A.1(c) is irrespective of the constant $r$, it suffices to study the magnitude of $\sum_{t=[\tau_{0}T-k_{T}]+1}^{[\tau _{0}T]}y_{t-1}^{2}$ in order to obtain the magnitude of $\sum_{t=[\hat{\tau}_{T}T]+1}^{[\tau _{0}T]}y_{t-1}^{2}$. Therefore, when $\hat{\tau}_{T}\leq \tau _{0}$, Lemma A.1(c) implies that
\begin{equation}  \label{A.6}
\sum_{t=[\tau_{0}T-k_{T}]+1}^{[\tau_{0}T]}y_{t-1}^{2}=O_{p}(k_{T}l(\eta_{T}))\cdot k_{T}=O_{p}(k_{T}^{2}l(\eta _{T})),
\end{equation}
which yields
\begin{equation}  \label{A.1}
\sum_{t=[\hat{\tau}_{T}T]+1}^{[\tau_{0}T]}y_{t-1}^{2}=O_{p}(k_{T}^{2}l(\eta_{T})).
\end{equation}
In addition, to obtain the magnitude of $\sum_{t=[\hat{\tau}_{T}T]+1}^{[\tau_{0}T]}y_{t-1}\varepsilon_{t}$, it suffices to study the magnitude of $\sum_{t=[\tau_{0}T-k_{T}]}^{[\tau_{0}T]}y_{t-1}\varepsilon_{t}$. By squaring $y_{t}=\beta _{1T}y_{t-1}+\varepsilon_{t}$ and summing over $t\in\{[\tau _{0}T-k_{T}]+1,\cdots ,[\tau_{0}T]\}$ we obtain
\begin{eqnarray}\label{A.0}
\sum_{t=[\tau _{0}T-k_{T}]+1}^{[\tau _{0}T]}y_{t-1}\varepsilon _{t} &=&\frac{1-\beta _{1T}^{2}}{2\beta _{1T}}\sum_{t=[\tau _{0}T-k_{T}]+1}^{[\tau_{0}T]}y_{t-1}^{2}-\frac{\beta _{1T}}{2}(y_{[\tau _{0}T-k_{T}]}^{2}-y_{[\tau_{0}T]}^{2})  \notag  \\
&&-\frac{1}{2\beta _{1T}}\sum_{t=[\tau _{0}T-k_{T}]+1}^{[\tau_{0}T]}\varepsilon _{t}^{2}.
\end{eqnarray}
It follows from (\ref{A.6}) and Lemma A.1(c) respectively that\label{A.2}
\begin{eqnarray}
\frac{1-\beta _{1T}^{2}}{2\beta _{1T}}\sum_{t=[\tau_{0}T-k_{T}]+1}^{[\tau _{0}T]}y_{t-1}^{2}=O_{p}(k_{T}l(\eta
_{T}))
\end{eqnarray}
and
\begin{eqnarray} \label{A.3}
\frac{\beta _{1T}}{2}(y_{[\tau _{0}T-k_{T}]}^{2}-y_{[\tau_{0}T]}^{2})=\frac{\beta _{1T}}{2}(y_{[\tau
_{0}T-k_{T}]}-y_{[\tau _{0}T]})(y_{[\tau _{0}T-k_{T}]}+y_{[\tau_{0}T]})=O_{p}(k_{T}l(\eta _{T})).
\end{eqnarray}
Moreover, by applying the Law of Large Numbers, we have
\begin{equation}  \label{A.4}
\frac{1}{2\beta _{1T}}\sum_{t=[\tau _{0}T-k_{T}]+1}^{[\tau_{0}T]}\varepsilon _{t}^{2}=O_{p}(k_{T}l(\eta _{T})).
\end{equation}
Combining (\ref{A.0})-(\ref{A.4}) leads to
\begin{equation*}
\sum_{t=[\tau _{0}T-k_{T}]+1}^{[\tau _{0}T]}y_{t-1}\varepsilon_{t}=O_{p}(k_{T}l(\eta _{T})),
\end{equation*}
yielding
\begin{equation}
\sum_{t=[\hat{\tau}_{T}T]+1}^{[\tau _{0}T]}y_{t-1}\varepsilon_{t}=O_{p}(k_{T}l(\eta _{T})).  \label{1.14}
\end{equation}
Similarly, if $\hat{\tau}_{T}>\tau _{0}$, we have
\begin{eqnarray*}
\sum_{t=[\tau _{0}T]+1}^{[\tau _{0}T+k_{T}]}y_{t-1}^{2} &=&\sum_{t=[\tau_{0}T]+1}^{[\tau_{0}T+k_{T}]}\Big(y_{[\tau_{0}T]}+\sum_{i=1}^{t-1-[\tau_{0}T]}\varepsilon _{\lbrack \tau _{0}T]+i}\Big)^{2} \\
&\leq &2\sum_{t=[\tau_{0}T]+1}^{[\tau_{0}T+k_{T}]}y_{[\tau_{0}T]}^{2}+2\sum_{t=[\tau _{0}T]+1}^{[\tau _{0}T+k_{T}]}\Big(\sum_{i=1}^{t-1-[\tau _{0}T]}\varepsilon _{\lbrack \tau_{0}T]+i}\Big)^{2} \\
&=&O_{p}(k_{T}^{2}l(\eta_{T}))+O_{p}(k_{T}^{2}l(\eta_{T})) \\
&=&O_{p}(k_{T}^{2}l(\eta_{T})),
\end{eqnarray*}
leading to
\begin{equation}  \label{A.5}
\sum_{t=[\tau_{0}T]+1}^{[\hat{\tau}_{T}T]}y_{t-1}^{2}=O_{p}(k_{T}^{2}l(\eta_{T})).
\end{equation}
In addition, by squaring $y_{t}=y_{t-1}+\varepsilon_{t}$ and summing over $t\in \{[\tau _{0}T]+1,\cdots ,[\tau _{0}T+k_{T}]\}$ we obtain
\begin{eqnarray*}
\sum_{t=[\tau_{0}T]+1}^{[\tau_{0}T+k_{T}]}y_{t-1}\varepsilon_{t} &=&\frac{1}{2}\left( y_{[\tau_{0}T+k_{T}]}^{2}-y_{[\tau_{0}T]}^{2}\right) -\frac{1}{2}\sum_{t=[\tau_{0}T]+1}^{[\tau_{0}T+k_{T}]}\varepsilon_{t}^{2} \\
&=&\frac{1}{2}\left( 2y_{[\tau_{0}T]}\sum_{t=[\tau _{0}T]+1}^{[\tau_{0}T+k_{T}]}\varepsilon_{t}+\Big(\sum_{t=[\tau_{0}T]+1}^{[\tau_{0}T+k_{T}]}\varepsilon _{t}\Big)^{2}\right)-\frac{1}{2}\sum_{t=[\tau_{0}T]+1}^{[\tau _{0}T+k_{T}]}\varepsilon_{t}^{2}.
\end{eqnarray*}
Similarly to the derivation of (\ref{1.14}), we have
\begin{equation}  \label{1.16}
\sum_{t=[\tau _{0}T]+1}^{[\hat{\tau}_{T}T]}y_{t-1}\varepsilon_{t}=O_{p}(k_{T}l(\eta _{T})).
\end{equation}
$\hfill \Box $\newline

\bigskip


\noindent \textbf{Proof of Lemma B.1}. \ This lemma can easily be proved by using the standard arguments in the unit root model with finite variance and by
applying the truncation technique (A.1). The details are omitted. $\hfill \Box $\newline

\noindent \textbf{Proof of Lemma B.2.} \ The proof can be completed by following the proof of part (c) of Lemma A.1. The details are therefore omitted. $\hfill \Box $\newline

\noindent \textbf{Proof of Lemma B.3.} \ To prove part (a), we note that the following decomposition holds,
\begin{eqnarray}\label{2.12}
&&\frac{1}{\sqrt{Tk_{T}}l(\eta _{T})}\sum_{t=[\tau_{0}T]+1}^{T}y_{t-1}\varepsilon _{t}  \notag  \\
&=&\frac{1}{\sqrt{Tk_{T}}l(\eta _{T})}\sum_{t=[\tau _{0}T]+1}^{T}\Big(\beta_{2T}^{t-[\tau _{0}T]-1}y_{[\tau _{0}T]}+\sum_{i=[\tau _{0}T]+1}^{t-1}\beta
_{2T}^{t-i-1}\varepsilon _{i}\Big)\varepsilon _{t}  \notag \\
&=&\frac{1}{\sqrt{Tk_{T}}l(\eta _{T})}\Big(\Big(y_{0}+\sum_{j=1}^{[\tau_{0}T]}\varepsilon _{j}\Big)\sum_{t=[\tau _{0}T]+1}^{T}\beta _{2T}^{t-[\tau
_{0}T]-1}\varepsilon _{t}+\sum_{t=[\tau _{0}T]+1}^{T}\varepsilon_{t}\sum_{i=[\tau _{0}T]+1}^{t-1}\beta _{2T}^{t-i-1}\varepsilon _{i}\Big)\notag \\
&:=&I_{1}+I_{2},
\end{eqnarray}
where
\begin{eqnarray*}
I_{1} &=&\frac{1}{\sqrt{Tk_{T}}l(\eta _{T})}\Big(y_{0}+\sum_{j=1}^{[\tau_{0}T]}\varepsilon _{j}\Big)\sum_{t=[\tau _{0}T]+1}^{T}\beta_{2T}^{t-[\tau
_{0}T]-1}\varepsilon _{t} \\
&=&\frac{1+o_{p}(1)}{\sqrt{Tk_{T}}l(\eta _{T})}\sum_{j=1}^{[\tau_{0}T]}\varepsilon _{j}\sum_{t=[\tau _{0}T]+1}^{T}\beta _{2T}^{t-[\tau_{0}T]-1}\varepsilon _{t} \\
&=&\frac{1+o_{p}(1)}{\sqrt{Tk_{T}}l(\eta _{T})}\sum_{j=1}^{[\tau_{0}T]}\varepsilon _{j}^{(1)}\sum_{t=[\tau _{0}T]+1}^{T}\beta _{2T}^{t-[\tau_{0}T]-1}\varepsilon _{t}^{(1)}
\end{eqnarray*}
and
\begin{eqnarray*}
I_{2} &=&\frac{1}{\sqrt{Tk_{T}}l(\eta _{T})}\sum_{t=[\tau_{0}T]+1}^{T}\varepsilon _{t}\sum_{i=[\tau _{0}T]+1}^{t-1}\beta_{2T}^{t-i-1}\varepsilon _{i} \\
&=&\frac{1+o_{p}(1)}{\sqrt{Tk_{T}}l(\eta _{T})}\sum_{t=[\tau_{0}T]+1}^{T}\varepsilon _{t}^{(1)}\sum_{i=[\tau _{0}T]+1}^{t-1}\beta_{2T}^{t-i-1}\varepsilon_{i}^{(1)}
\end{eqnarray*}
by assumptions C1-C3 and Lemma 1 in Cs\"{o}rg\H{o} et al. (2003).

For the term $I_{1}$, it is obvious that
\begin{equation}  \label{2.5}
\phi_j=\frac{1}{\sqrt{Tk_T}}\frac{\varepsilon _{j}^{(1)}}{\sqrt{l(\eta _{T})}}\sum_{t=[\tau_{0}T]+1}^{T}\beta _{2T}^{t-[\tau _{0}T]-1}\frac{\varepsilon _{t}^{(1)}}{\sqrt{l(\eta _{T})}},~~j=1, \cdots, [\tau _{0}T]
\end{equation}
is a sequence of martingale difference with respect to the filtration $\mathfrak{F}_j=\sigma(\varepsilon_1,\cdots,\varepsilon_j)$ with
\begin{equation*}
\sum_{j=1}^{[\tau_0 T]}E(\phi_j^2|\mathfrak{F}_{j-1})=\frac{1}{Tk_T}\sum_{t=[\tau_{0}T]+1}^{T}\beta _{2T}^{2(t-[\tau _{0}T]-1)}\cdot (1+o(1))=\frac{\tau_0}{2c}\cdot (1+o(1)).
\end{equation*}
In addition, for any $\delta>0$, we have
\begin{eqnarray*}
\sum_{j=1}^{[\tau_0 T]}E(\phi_j^2I\{|\phi_j|>\delta\}|\mathfrak{F}_{j-1})&=&[\tau_0 T]E(\phi_1^2I\{|\phi_1|>\delta\})\non
&=&\frac{[\tau_0 T]}{T}E\left((\sqrt{T}\phi_1)^2I\{|\sqrt{T}\phi_1|>\delta\sqrt{T}\}\right)\non
&\rightarrow&0
\end{eqnarray*}
since $E(\sqrt{T}\phi_1)^2<\infty$. Hence, the Lindeberg condition is verified. Then, applying the central limit theorem for martingale difference sequences, we have
\begin{equation}  \label{2.11}
I_{1}\Rightarrow N(0,\frac{\tau_{0}}{2c}).
\end{equation}

For the term $I_{2}$, it is easy to see that
\begin{equation}\label{2.8}
\varphi_t=\frac{1}{\sqrt{Tk_T}}\frac{\varepsilon _{t}^{(1)}}{\sqrt{l(\eta _{T})}}\sum_{i=[\tau_{0}T]+1}^{t-1}\beta _{2T}^{t-i-1}\frac{\varepsilon _{i}^{(1)}}{\sqrt{l(\eta_{T})}},~~t=[\tau _{0}T]+1,\cdots, T
\end{equation}
is a sequence of martingale difference with respect to the filtration $\mathfrak{F}_{t}=\sigma (\varepsilon _{\lbrack \tau _{0}T]},\cdots,\varepsilon_{t})$ with %\begin{equation*}
%E\Big(\frac{1}{\sqrt{Tk_T}}\frac{\varepsilon _{t}^{(1)}}{\sqrt{l(\eta _{T})}}\sum_{i=[\tau_{0}T]+1}^{t-1}\beta _{2T}^{t-i-1}\frac{\varepsilon %_{i}^{(1)}}{\sqrt{l(\eta_{T})}}|\mathfrak{F}_{t-1}\Big)=0
%\end{equation*}
%and
\begin{equation*}
\sum_{t=[\tau _{0}T]+1}^{T}E(\varphi_t^2|\mathfrak{F}_{t-1})=\frac{1-\tau _{0}}{2c}(1+o_p(1)).
\end{equation*}
In addition, by the similar arguments in pages 203-204 in Huang et al. (2014), it is easy to verify that the Lindeberg condition
\begin{eqnarray*}
\sum_{t=[\tau _{0}T]+1}^{T}E(\varphi_t^2I\{|\varphi_t|>\delta\}|\mathfrak{F}_{t-1})=o_p(1),~~{\rm for~ any}~ \delta>0
\end{eqnarray*}
holds. Hence, applying the central limit theorem for martingale difference sequences leads to
\begin{equation}\label{2.6}
I_{2}\Rightarrow N(0,\frac{1-\tau _{0}}{2c}).
\end{equation}
Note that the two sequences (\ref{2.5}) and (\ref{2.8}) are independent. The proof of part (a) is then completed by combining (\ref{2.12}), (\ref{2.11}) and (\ref{2.6}).

To prove part (b), by squaring $y_{t}=(1-c/k_{T})y_{t-1}+\varepsilon_{t}$ and summing over $t\in \{[\tau _{0}T]+1,\cdots ,T\}$, we obtain
\begin{equation*}
\sum_{t=[\tau_{0}T]+1}^{T}y_{t}^{2}=(1-\frac{c}{k_{T}})^{2}\sum_{t=[\tau_{0}T]+1}^{T}y_{t-1}^{2}+\sum_{t=[\tau _{0}T]+1}^{T}\varepsilon _{t}^{2}+2(1-\frac{c}{k_{T}})\sum_{t=[\tau_{0}T]+1}^{T}y_{t-1}\varepsilon_{t}.
\end{equation*}
This implies that
\begin{equation}
(\frac{2c}{k_{T}}-\frac{c^{2}}{k_{T}^{2}})\sum_{t=[\tau_{0}T]+1}^{T}y_{t-1}^{2}=y_{[\tau _{0}T]}^{2}-y_{T}^{2}+\sum_{t=[\tau_{0}T]+1}^{T}\varepsilon _{t}^{2}+2(1-\frac{c}{k_{T}})\sum_{t=[\tau_{0}T]+1}^{T}y_{t-1}\varepsilon_{t}.  \label{2.16}
\end{equation}
Note that
\begin{equation}\label{2.18}
\frac{y_{[\tau _{0}T]}}{\sqrt{Tl(\eta _{T})}}\Rightarrow W(\tau _{0})
\end{equation}
since $\beta_{1}=1$, it follows from Lemma B.2 and assumption C2 that
\begin{equation}\label{2.17}
\frac{y_{T}}{\sqrt{Tl(\eta _{T})}}=\frac{\beta _{2T}^{T-[\tau_{0}T]}y_{[\tau _{0}T]}+\sum_{i=[\tau _{0}T]+1}^{T}\beta_{2T}^{T-i}\varepsilon _{i}}{\sqrt{Tl(\eta _{T})}}=o_{p}(1).
\end{equation}
In addition,
\begin{equation}\label{2.19}
2(1-\frac{c}{k_{T}})\sum_{t=[\tau _{0}T]+1}^{T}y_{t-1}\varepsilon_{t}=o_{p}(Tl(\eta _{T}))
\end{equation}
by part (a). Then, combining (\ref{2.16})-(\ref{2.19}) and (\ref{1.11}) together yields
\begin{equation*}
\frac{1}{Tk_{T}l(\eta _{T})}\sum_{t=[\tau_{0}T]+1}^{T}y_{t-1}^{2}\Rightarrow \frac{1}{2c}(W^{2}(\tau _{0})+1-\tau_{0}).
\end{equation*}

Note that the random part (i.e., $W^{2}(\tau _{0})$) of the limiting distribution of $\frac{1}{Tk_{T}l(\eta _{T})}\sum_{t=[\tau_{0}T]+1}^{T}y_{t-1}^{2}$ is due to the limiting behavior of $(\sum_{i=1}^{[\tau _{0}T]}\varepsilon _{i})^{2}$, while the limiting distribution of $\frac{1}{\sqrt{Tk_{T}}l(\eta _{T})}\sum_{t=[%
\tau_{0}T]+1}^{T}y_{t-1}\varepsilon _{t}$ is determined by the limiting behavior of
\begin{equation*}
\sum_{j=1}^{[\tau_{0}T]}\varepsilon_{j}\sum_{t=[\tau_{0}T]+1}^{T}\beta_{2T}^{t-[\tau_{0}T]-1}\varepsilon_{t}~~\mathrm{and}~~\sum_{t=[\tau_{0}T]+1}^{T}\varepsilon_{t}\sum_{i=[\tau_{0}T]+1}^{t-1}\beta_{2T}^{t-i-1}\varepsilon_{i}.
\end{equation*}
Note also that $\sum_{i=1}^{[\tau _{0}T]}\varepsilon _{i}$ and $\sum_{t=[\tau_{0}T]+1}^{T}\varepsilon_{t}\sum_{i=[\tau_{0}T]+1}^{t-1}\beta_{2T}^{t-i-1}\varepsilon _{i}$ are independent and $\sum_{i=1}^{[\tau _{0}T]}\varepsilon_{i}$ and $\sum_{j=1}^{[\tau_{0}T]}\varepsilon _{j}\sum_{t=[\tau _{0}T]+1}^{T}\beta_{2T}^{t-[\tau_{0}T]-1}\varepsilon _{t}$ are uncorrelated. Hence, the two limiting distributions in part (a) and part (b) are independent. Therefore, part (a) and part (b) hold jointly. $\hfill \Box $\newline



\noindent \textbf{Proof of Lemma B.4.} \ By applying Lemmas B.1 and B.3, it is trivial that $A_{1}=O_{p}(1/\sqrt{Tk_{T}})=o_{p}(1/k_{T})$ and $A_{4}=O_{p}(1/T)=o_{p}(1/k_{T})$. To find the order of the term $A_{2}$, note that since $\beta _{1}=1$, it is not difficult to see that
\begin{eqnarray*}
\sum_{t=m+1}^{[\tau_{0}T]}y_{t-1}\varepsilon_{t}&=&y_{0}\sum_{t=m+1}^{[\tau_{0}T]}\varepsilon_{t}+\sum_{t=m+1}^{[\tau
_{0}T]}\left( \sum_{i=1}^{t-1}\varepsilon _{i}\right) \varepsilon _{t} \\
&=&\left( y_{0}\sum_{t=m+1}^{[\tau_{0}T]}\varepsilon_{t}^{(1)}+\sum_{t=m+1}^{[\tau_{0}T]}\left( \sum_{i=1}^{t-1}\varepsilon
_{i}^{(1)}\right) \varepsilon_{t}^{(1)}\right) \cdot (1+o_{p}(1)) \\
&=&o_{p}\left( \sqrt{T([\tau_{0}T]-m)l(\eta_{T})}\right) +O_{p}\left(\sqrt{\sum_{t=m+1}^{[\tau_{0}T]}E\left( \sum_{i=1}^{t-1}\varepsilon
_{i}^{(1)}\right) ^{2}l(\eta _{T})}\right) \\
&=&o_{p}\left( \sqrt{T([\tau_{0}T]-m)l(\eta_{T})}\right) +O_{p}\left(\sqrt{([\tau _{0}T]-m)([\tau_{0}T]+m)}l(\eta_{T})\right) \\
&=&O_{p}\left( \sqrt{([\tau_{0}T]-m)([\tau_{0}T]+m)}l(\eta_{T})\right)
\end{eqnarray*}
and
\begin{eqnarray*}
\sum_{t=m+1}^{[\tau_{0}T]}y_{t-1}^{2} &=&T([\tau_{0}T]-m)l(\eta_{T})\cdot\frac{1}{[\tau _{0}T]-m}\sum_{t=m+1}^{[\tau _{0}T]}\left( \frac{y_{t-1}}{\sqrt{Tl(\eta_{T})}}\right) ^{2} \\
&=&O_{p}(T([\tau_{0}T]-m)l(\eta_{T})).
\end{eqnarray*}
Consequently, we have
\begin{equation}\label{B.4}
A_{2}=O_{p}\left( \sup_{m\in D_{1T}}\frac{\sqrt{([\tau_{0}T]-m)([\tau_{0}T]+m)}}{T([\tau _{0}T]-m)}\right) \leq O_{p}\left( \frac{1}{\sqrt{TM_{T}}}\right)=o_{p}(1/k_{T}).
\end{equation}
To find the order of the term $A_{5}$, note that for any small $0<\delta <1$, there exist two large constants $N_{1}=N_{1}(\delta )$ such that $\beta
_{2T}^{k_{T}/N_{1}}\rightarrow e^{-c/N_{1}}>1-\delta $ and $N_{2}=N_{2}(\delta )$ such that $\beta _{2T}^{N_{2}k_{T}}\rightarrow e^{-cN_{2}}<\delta $. We then divide the set $\{M_{T}+1,\cdots,T-[\tau_{0}T]\}$, which is the domain of $m-[\tau _{0}T]$ when $m\in D_{2T}$, into three subsets: $\{M_{T}+1,\cdots ,[k_{T}/N_{1}]\}$, $\{[k_{T}/N_{1}]+1,\cdots ,[N_{2}k_{T}]\}$ and $\{[N_{2}k_{T}]+1,\cdots,T-[\tau _{0}T]\}$. Note that
\begin{equation*}
\sum_{t=[\tau _{0}T]+1}^{m}y_{t-1}\varepsilon _{t}=\sum_{t=[\tau_{0}T]+1}^{m}\left( \beta _{2T}^{t-1-[\tau _{0}T]}y_{[\tau
_{0}T]}+\sum_{i=[\tau _{0}T]+1}^{t-1}\beta _{2T}^{t-1-i}\varepsilon_{i}\right) \varepsilon _{t},
\end{equation*}
\begin{eqnarray*}
&&y_{[\tau _{0}T]}\sum_{t=[\tau _{0}T]+1}^{m}\beta _{2T}^{t-1-[\tau_{0}T]}\varepsilon _{t} \\
&=&O_{p}(\sqrt{Tl(\eta _{T})})\cdot O_{p}\left( \sqrt{k_{T}(1-\beta_{2T}^{2(m-[\tau _{0}T])})l(\eta _{T})}\right) \\
&=&O_{p}\left( \sqrt{Tk_{T}(1-\beta _{2T}^{2(m-[\tau _{0}T])})}l(\eta_{T})\right) \\
&=&\left\{
\begin{array}{ll}
O_{p}\left( \sqrt{T(m-[\tau _{0}T])}l(\eta _{T})\right) ,~ & \mbox{if}~M_{T}<m-[\tau _{0}T]\leq \lbrack k_{T}/N_{1}] \\
O_{p}\left( \sqrt{Tk_{T}}l(\eta _{T})\right) ,~ & \mbox{if}~[k_{T}/N_{1}]<m-[\tau _{0}T]\leq \lbrack N_{2}k_{T}] \\
O_{p}\left( \sqrt{Tk_{T}}l(\eta _{T})\right) ,~ & \mbox{if}~[N_{2}k_{T}]<m-[\tau _{0}T]\leq T-[\tau _{0}T]%
\end{array}
\right.
\end{eqnarray*}
since
\begin{equation*}
\left( 1-(1-c/k_{T})^{a_{T}}\right) =O(a_{T}/k_{T})~~\mbox{if}~~a_{T}=o(k_{T})
\end{equation*}
and
\begin{eqnarray*}
&&\sum_{t=[\tau _{0}T]+1}^{m}\left( \sum_{i=[\tau _{0}T]+1}^{t-1}\beta_{2T}^{t-1-i}\varepsilon _{i}\right) \varepsilon _{t} \\
&=&O_{p}\left( \sqrt{\sum_{t=[\tau _{0}T]+1}^{m}\sum_{i=[\tau _{0}T]+1}^{t-1}\beta_{2T}^{2(t-1-i)}}l(\eta_T)\right) \\
&=&O_{p}\left( \sqrt{k_{T}(m-[\tau _{0}T])-k_{T}^{2}(1-\beta_{2T}^{2(m-[\tau _{0}T])})}l(\eta _{T})\right) \\
&=&\left\{
\begin{array}{ll}
O_{p}\left( \sqrt{k_{T}(m-[\tau _{0}T])}l(\eta _{T})\right) ,~ & \mbox{if}~M_{T}<m-[\tau _{0}T]\leq \lbrack k_{T}/N_{1}] \\
O_{p}\left( k_{T}l(\eta _{T})\right) ,~ & \mbox{if}~[k_{T}/N_{1}]<m-[\tau_{0}T]\leq \lbrack N_{2}k_{T}] \\
O_{p}\left( \sqrt{k_{T}(m-[\tau _{0}T])}l(\eta _{T})\right) ,~ & \mbox{if}~[N_{2}k_{T}]<m-[\tau _{0}T]\leq T-[\tau _{0}T]%
\end{array}.
\right.
\end{eqnarray*}
We then have
\begin{eqnarray}
&&\sum_{t=[\tau _{0}T]+1}^{m}y_{t-1}\varepsilon_{t}  \notag  \label{B.8} \\
&=&\left\{
\begin{array}{ll}
O_{p}\left( \sqrt{T(m-[\tau _{0}T])}l(\eta _{T})\right) ,~ & \mbox{if}~M_{T}<m-[\tau _{0}T]\leq \lbrack k_{T}/N_{1}] \\
O_{p}\left( \sqrt{Tk_{T}}l(\eta _{T})\right) ,~ & \mbox{if}~[k_{T}/N_{1}]<m-[\tau _{0}T]\leq \lbrack N_{2}k_{T}] \\
O_{p}\left( \sqrt{Tk_{T}}l(\eta _{T})\right) ,~ & \mbox{if}~[N_{2}k_{T}]<m-[\tau _{0}T]\leq T-[\tau _{0}T]%
\end{array}
.\right.
\end{eqnarray}
Moreover, we also have
\begin{eqnarray}\label{B.23}
&&\sum_{t=[\tau_{0}T]+1}^{m}y_{t-1}^{2}  \notag  \\
&=&-\frac{1}{1-\beta_{2T}^{2}}(y_{m}^{2}-y_{[\tau _{0}T]}^{2})+\frac{1}{1-\beta _{2T}^{2}}\sum_{t=[\tau _{0}T]+1}^{m}\varepsilon _{t}^{2}+\frac{2\beta _{2T}}{1-\beta_{2T}^{2}}\sum_{t=[\tau_{0}T]+1}^{m}y_{t-1}\varepsilon_{t},
\end{eqnarray}
\begin{eqnarray}\label{B.24}
&&\frac{2\beta_{2T}}{1-\beta _{2T}^{2}}\sum_{t=[\tau_{0}T]+1}^{m}y_{t-1}\varepsilon _{t}  \notag  \\
&=&\left\{
\begin{array}{ll}
O_{p}\left( k_{T}\sqrt{T(m-[\tau _{0}T])}l(\eta _{T})\right) ,~ & \mbox{if}~M_{T}<m-[\tau _{0}T]\leq \lbrack k_{T}/N_{1}] \\
O_{p}\left( \sqrt{T}k_{T}^{3/2}l(\eta _{T})\right) ,~ & \mbox{if}~[k_{T}/N_{1}]<m-[\tau _{0}T]\leq \lbrack N_{2}k_{T}] \\
O_{p}\left( \sqrt{T}k_{T}^{3/2}l(\eta _{T})\right) ,~ & \mbox{if}~[N_{2}k_{T}]<m-[\tau _{0}T]\leq T-[\tau _{0}T]%
\end{array}
\right.
\end{eqnarray}
by (\ref{B.8}),
\begin{eqnarray}\notag  \label{B.25}
&&\frac{1}{1-\beta _{2T}^{2}}\sum_{t=[\tau _{0}T]+1}^{m}\varepsilon_{t}^{2}\\
&=&O_{p}\left( k_{T}(m-[\tau _{0}T])l(\eta _{T})\right)  \notag \\
&=&\left\{
\begin{array}{ll}
O_{p}\left( k_{T}(m-[\tau _{0}T])l(\eta _{T})\right) ,~ & \mbox{if}~M_{T}<m-[\tau _{0}T]\leq \lbrack k_{T}/N_{1}] \\
O_{p}\left( k_{T}^{2}l(\eta _{T})\right) ,~ & \mbox{if}~[k_{T}/N_{1}]<m-[\tau _{0}T]\leq \lbrack N_{2}k_{T}] \\
O_{p}\left( k_{T}(m-[\tau _{0}T])l(\eta _{T})\right) ,~ & \mbox{if}~[N_{2}k_{T}]<m-[\tau _{0}T]\leq T-[\tau _{0}T]
\end{array}
\right.
\end{eqnarray}
and
\begin{eqnarray}\label{B.26}
&&-\frac{1}{1-\beta_{2T}^{2}}(y_{m}^{2}-y_{[\tau_{0}T]}^{2})  \notag\\
&=&-\frac{1}{1-\beta _{2T}^{2}}(y_{m}+y_{[\tau _{0}T]})(y_{m}-y_{[\tau_{0}T]})  \notag \\
&=&-\frac{1}{1-\beta _{2T}^{2}}\left( \sum_{i=[\tau _{0}T]+1}^{m}\beta_{2T}^{m-i}\varepsilon _{i}+\left( \beta _{2T}^{m-[\tau _{0}T]}+1\right)
y_{[\tau _{0}T]}\right)  \notag \\
&&\cdot \left( \sum_{i=[\tau _{0}T]+1}^{m}\beta _{2T}^{m-i}\varepsilon_{i}+\left( \beta _{2T}^{m-[\tau _{0}T]}-1\right) y_{[\tau _{0}T]}\right)
\notag \\
&=&k_{T}\cdot \left( O_{p}\left( \sqrt{k_{T}(1-\beta _{2T}^{2(m-[\tau_{0}T])})l(\eta _{T})}\right) +O_{p}(\sqrt{Tl(\eta _{T})})\right)  \notag \\
&&\cdot \left( O_{p}\left( \sqrt{k_{T}(1-\beta _{2T}^{2(m-[\tau_{0}T])})l(\eta _{T})}\right) +O_{p}\left( (1-\beta _{2T}^{m-[\tau _{0}T]})%
\sqrt{Tl(\eta _{T})}\right) \right)  \notag \\
&=&O_{p}(k_{T}\sqrt{Tl(\eta _{T})})\cdot \left( O_{p}\left( \sqrt{k_{T}(1-\beta _{2T}^{2(m-[\tau _{0}T])})l(\eta _{T})}\right) +O_{p}\left(
(1-\beta _{2T}^{m-[\tau _{0}T]})\sqrt{Tl(\eta _{T})}\right) \right)  \notag\\
&=&\left\{
\begin{array}{ll}
O_{p}(k_{T}\sqrt{Tl(\eta _{T})})\cdot O_{p}\left( \frac{m-[\tau _{0}T]}{k_{T}}\sqrt{Tl(\eta _{T})}\right) ,~ & \mbox{if}~M_{T}<m-[\tau _{0}T]\leq \lbrack
k_{T}/N_{1}] \\
O_{p}(k_{T}\sqrt{Tl(\eta _{T})})\cdot O_{p}(\sqrt{Tl(\eta _{T})}),~ & \mbox{if}~[k_{T}/N_{1}]<m-[\tau _{0}T]\leq \lbrack N_{2}k_{T}] \\
O_{p}(k_{T}\sqrt{Tl(\eta _{T})})\cdot O_{p}(\sqrt{Tl(\eta _{T})}),~ & \mbox{if}~[N_{2}k_{T}]<m-[\tau _{0}T]\leq T-[\tau _{0}T]%
\end{array}
\right.  \notag \\
&=&\left\{
\begin{array}{ll}
O_{p}\left( T(m-[\tau _{0}T])l(\eta _{T})\right) ,~ & \mbox{if}~M_{T}<m-[\tau _{0}T]\leq \lbrack k_{T}/N_{1}] \\
O_{p}(Tk_{T}l(\eta _{T})),~ & \mbox{if}~[k_{T}/N_{1}]<m-[\tau _{0}T]\leq\lbrack N_{2}k_{T}] \\
O_{p}(Tk_{T}l(\eta _{T})),~ & \mbox{if}~[N_{2}k_{T}]<m-[\tau _{0}T]\leq T-[\tau _{0}T]%
\end{array}
.\right.
\end{eqnarray}
Combining (\ref{B.23})-(\ref{B.26}), we have
\begin{eqnarray}\label{B.27}
&&\sum_{t=[\tau_{0}T]+1}^{m}y_{t-1}^{2}  \notag  \\
&=&\left\{
\begin{array}{ll}
O_{p}\left( \max \left\{ k_{T}\sqrt{T(m-[\tau _{0}T])}l(\eta _{T}),T(m-[\tau_{0}T])l(\eta _{T})\right\} \right) , &  \\
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\mbox{if}~M_{T}<m-[\tau _{0}T]\leq \lbrack k_{T}/N_{1}] &  \\
O_{p}(Tk_{T}l(\eta _{T})),~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\mbox{if}~[k_{T}/N_{1}]<m-[\tau _{0}T]\leq \lbrack N_{2}k_{T}] &  \\
O_{p}\left( \max \left\{ k_{T}(m-[\tau _{0}T])l(\eta _{T}),Tk_{T}l(\eta_{T})\right\} \right) ,\quad \mbox{if}~[N_{2}k_{T}]<m-[\tau _{0}T]\leq T-[\tau _{0}T] &
\end{array}
\right.  \notag \\
&=&\left\{
\begin{array}{ll}
O_{p}\left( T(m-[\tau _{0}T])l(\eta _{T})\right) ,~ & \mbox{if}~M_{T}<m-[\tau _{0}T]\leq \lbrack k_{T}/N_{1}] \\
O_{p}(Tk_{T}l(\eta _{T})),~ & \mbox{if}~[k_{T}/N_{1}]<m-[\tau _{0}T]\leq \lbrack N_{2}k_{T}] \\
O_{p}(Tk_{T}l(\eta _{T})),~ & \mbox{if}~[N_{2}k_{T}]<m-[\tau _{0}T]\leq T-[\tau _{0}T]%
\end{array}
.\right.  \notag \\
&&
\end{eqnarray}
Hence, it follows from (\ref{B.8}) and (\ref{B.27}) that, when $M_{T}<m-[\tau _{0}T]\leq \lbrack k_{T}/N_{1}]$, we have
\begin{equation*}
\frac{\sum_{t=[\tau_{0}T]+1}^{m}y_{t-1}\varepsilon _{t}}{\sum_{t=[\tau_{0}T]+1}^{m}y_{t-1}^{2}}=O_{p}\left( \frac{\sqrt{T(m-[\tau _{0}T])}l(\eta
_{T})}{T(m-[\tau_{0}T])l(\eta_{T})}\right) \leq O_{p}\left( \frac{1}{\sqrt{TM_{T}}}\right) =o_{p}(1/k_{T})
\end{equation*}
since $k_T=O(\sqrt{T})$ and $M_T\rightarrow \infty$. When $[k_{T}/N_{1}]<m-[\tau _{0}T]\leq\lbrack N_{2}k_{T}]$ or $[N_{2}k_{T}]<m-[\tau _{0}T]\leq T-[\tau _{0}T]$, we
have
\begin{equation*}
\frac{\sum_{t=[\tau _{0}T]+1}^{m}y_{t-1}\varepsilon _{t}}{\sum_{t=[\tau_{0}T]+1}^{m}y_{t-1}^{2}}=O_{p}\left( \frac{\sqrt{Tk_{T}}l(\eta _{T})}{Tk_{T}l(\eta _{T})}\right) =o_{p}(1/k_{T}).
\end{equation*}
Using the above arguments, we have
\begin{equation*}
A_{5}=\sup_{m\in D_{2T}}\frac{\sum_{t=[\tau _{0}T]+1}^{m}y_{t-1}\varepsilon_{t}}{\sum_{t=[\tau _{0}T]+1}^{m}y_{t-1}^{2}}=o_{p}(1/k_{T}).
\end{equation*}
In addition, we have
\begin{eqnarray*}
A_{3} &=&\sup_{m\in D_{1T}}\Big|\frac{\sum_{t=m+1}^{T}y_{t-1}^{2}}{\sum_{t=[\tau _{0}T]+1}^{T}y_{t-1}^{2}\sum_{t=m+1}^{[\tau _{0}T]}y_{t-1}^{2}}\Lambda _{T}(\frac{m}{T})\Big| \\
&\leq &\bigg(\frac{1}{\sum_{t=[\tau _{0}T]+1}^{T}y_{t-1}^{2}}+\frac{1}{\sum_{t=[\tau _{0}T]-M_{T}}^{[\tau _{0}T]}y_{t-1}^{2}}\bigg)\bigg(\sup_{m\in
D_{1T}}\Big|\frac{(\sum_{t=1}^{[\tau _{0}T]}y_{t-1}\varepsilon _{t})^{2}}{\sum_{t=1}^{[\tau _{0}T]}y_{t-1}^{2}}-\frac{(\sum_{t=1}^{m}y_{t-1}\varepsilon _{t})^{2}}{\sum_{t=1}^{m}y_{t-1}^{2}}\Big| \\
&&~~~~+\sup_{m\in D_{1T}}\Big|\frac{(\sum_{t=[\tau
_{0}T]+1}^{T}y_{t-1}\varepsilon _{t})^{2}}{\sum_{t=[\tau_{0}T]+1}^{T}y_{t-1}^{2}}-\frac{(\sum_{t=m+1}^{T}y_{t-1}\varepsilon _{t})^{2}}{\sum_{t=m+1}^{T}y_{t-1}^{2}}\Big|\bigg) \\
&=&\Big(O_{p}\big(\frac{1}{Tk_{T}l(\eta _{T})}\big)+O_{p}\big(\frac{1}{TM_{T}l(\eta _{T})}\big)\Big)\cdot O_{p}(l(\eta _{T})) \\
&=&o_{p}(1/k_{T}^{2}),
\end{eqnarray*}
and
\begin{eqnarray*}
A_{6} &=&\sup_{m\in D_{2T}}\Big|\frac{\sum_{t=1}^{m}y_{t-1}^{2}}{\sum_{t=[\tau _{0}T]+1}^{m}y_{t-1}^{2}\sum_{t=1}^{[\tau _{0}T]}y_{t-1}^{2}}\Lambda _{T}(\frac{m}{T})\Big| \\
&\leq &\bigg(\frac{1}{\sum_{t=1}^{[\tau _{0}T]}y_{t-1}^{2}}+\frac{1}{\sum_{t=[\tau _{0}T]+1}^{[\tau _{0}T]+M_{T}}y_{t-1}^{2}}\bigg)\bigg(\sup_{m\in D_{2T}}\Big|\frac{(\sum_{t=1}^{[\tau _{0}T]}y_{t-1}\varepsilon
_{t})^{2}}{\sum_{t=1}^{[\tau _{0}T]}y_{t-1}^{2}}-\frac{(\sum_{t=1}^{m}y_{t-1}\varepsilon _{t})^{2}}{\sum_{t=1}^{m}y_{t-1}^{2}}\Big|\\
&&~~~~+\sup_{m\in D_{2T}}\Big|\frac{(\sum_{t=[\tau_{0}T]+1}^{T}y_{t-1}\varepsilon_{t})^{2}}{\sum_{t=[\tau_{0}T]+1}^{T}y_{t-1}^{2}}-\frac{(\sum_{t=m+1}^{T}y_{t-1}\varepsilon_{t})^{2}}{\sum_{t=m+1}^{T}y_{t-1}^{2}}\Big|\bigg) \\
&\leq &\Big(O_{p}(\frac{1}{T^{2}l(\eta _{T})})+O_{p}(\frac{1}{TM_{T}l(\eta_{T})})\Big)\cdot O_{p}(l(\eta _{T})) \\
&=&o_{p}(1/k_{T}^{2}).
\end{eqnarray*}
The proofs are complete. $\hfill \Box $\newline


\noindent \textbf{Proof of Lemma B.5.} \ To prove part (a), using equation (B.2) in Chong (2001), we have
\begin{eqnarray*}
&&RSS_{T}(\tau _{0}-\frac{m}{T})-RSS_{T}(\tau_{0}) \\
&=&2(\beta_{2T}-\beta_{1})\bigg(\frac{\sum_{t=[\tau _{0}T]-m+1}^{[\tau_{0}T]}y_{t-1}^{2}\sum_{t=[\tau_{0}T]+1}^{T}y_{t-1}\varepsilon_{t}}{\sum_{t=[\tau _{0}T]-m+1}^{T}y_{t-1}^{2}}-\frac{\sum_{t=[\tau_{0}T]+1}^{T}y_{t-1}^{2}\sum_{t=[\tau _{0}T]-m+1}^{[\tau_{0}T]}y_{t-1}\varepsilon _{t}}{\sum_{t=[\tau _{0}T]-m+1}^{T}y_{t-1}^{2}}\bigg) \\
&&~+(\beta _{2T}-\beta _{1})^{2}\frac{\sum_{t=[\tau_{0}T]+1}^{T}y_{t-1}^{2}\sum_{t=[\tau _{0}T]-m+1}^{[\tau _{0}T]}y_{t-1}^{2}}{\sum_{t=[\tau_{0}T]-m+1}^{T}y_{t-1}^{2}}+\Lambda _{T}(\tau _{0}-\frac{m}{T})
\end{eqnarray*}
where
\begin{eqnarray*}
&&\Lambda _{T}(\tau _{0}-\frac{m}{T}) \\
&=&\frac{\Big(\sum_{t=1}^{[\tau _{0}T]}y_{t-1}\varepsilon _{t}\Big)^{2}}{\sum_{t=1}^{[\tau _{0}T]}y_{t-1}^{2}}-\frac{\Big(\sum_{t=1}^{[\tau_{0}T]-m}y_{t-1}\varepsilon _{t}\Big)^{2}}{\sum_{t=1}^{[\tau_{0}T]-m}y_{t-1}^{2}}+\frac{\Big(\sum_{t=[\tau
_{0}T]+1}^{T}y_{t-1}\varepsilon _{t}\Big)^{2}}{\sum_{t=[\tau_{0}T]+1}^{T}y_{t-1}^{2}}-\frac{\Big(\sum_{t=[\tau_{0}T]-m+1}^{T}y_{t-1}\varepsilon _{t}\Big)^{2}}{\sum_{t=[\tau_{0}T]-m+1}^{T}y_{t-1}^{2}}.
\end{eqnarray*}
Obviously, $\Lambda_{T}(\tau_{0}-\frac{m}{T})=O_{p}(l(\eta_{T}))$. Then it follows that
\begin{eqnarray*}
&&\frac{k_{T}^{2}}{Tl(\eta _{T})}\Big(RSS_{T}(\tau _{0}-\frac{m}{T})-RSS_{T}(\tau _{0})\Big) \\
&=&\frac{k_{T}^{2}}{Tl(\eta _{T})}\cdot \Big(-\frac{2c}{k_{T}}\Big)\cdot \bigg(\frac{O_{p}(Tl(\eta _{T}))O_{p}(\sqrt{Tk_{T}}l(\eta _{T}))}{O_{p}(Tl(\eta _{T}))+O_{p}(Tk_{T}l(\eta _{T}))}+\frac{O_{p}(Tk_{T}l(\eta_{T}))O_{p}(\sqrt{T}l(\eta _{T}))}{O_{p}(Tl(\eta
_{T}))+O_{p}(Tk_{T}l(\eta_{T}))}\bigg) \\
&&+\frac{k_{T}^{2}}{Tl(\eta _{T})}\cdot \frac{c^{2}}{k_{T}^{2}}\cdot \frac{O_{p}(Tk_{T}l(\eta _{T}))}{O_{p}(Tl(\eta _{T}))+O_{p}(Tk_{T}l(\eta _{T}))}\sum_{[\tau _{0}T]-m+1}^{[\tau _{0}T]}y_{t-1}^{2}+o_{p}(1) \\
&=&\frac{c^{2}(1+o_{p}(1))}{Tl(\eta _{T})}\sum_{[\tau_{0}T]-m+1}^{[\tau_{0}T]}y_{t-1}^{2}+o_{p}(1) \\
&\Rightarrow &c^{2}m W^{2}(\tau_{0})
\end{eqnarray*}
by Lemma B.3, $k_{T}=o(\sqrt{T})$ and the observation
\begin{equation*}
\left\vert \sum_{t=[\tau _{0}T]-m+1}^{[\tau _{0}T]}y_{t-1}\varepsilon_{t}\right\vert \leq \sqrt{\sum_{t=[\tau _{0}T]-m+1}^{[\tau_{0}T]}\varepsilon _{t}^{2}}\cdot \sqrt{\sum_{t=[\tau_{0}T]-m+1}^{[\tau_0 T]}y_{t-1}^{2}}=O_{p}(\sqrt{T}l(\eta _{T})).
\end{equation*}

To prove part (b), using equation (B.4) in Chong (2001), we have
\begin{eqnarray*}
&&RSS_{T}(\tau _{0}+\frac{m}{T})-RSS_{T}(\tau _{0}) \\
&=&2(\beta _{2T}-\beta _{1})\bigg(\frac{\sum_{t=1}^{[\tau_{0}T]}y_{t-1}^{2}\sum_{t=[\tau _{0}T]+1}^{[\tau _{0}T]+m}y_{t-1}\varepsilon
_{t}}{\sum_{t=1}^{[\tau _{0}T]+m}y_{t-1}^{2}}-\frac{\sum_{t=[\tau_{0}T]+1}^{[\tau _{0}T]+m}y_{t-1}^{2}\sum_{t=1}^{[\tau_{0}T]}y_{t-1}\varepsilon _{t}}{\sum_{t=1}^{[\tau _{0}T]+m}y_{t-1}^{2}}\bigg)\\
&&~+(\beta _{2T}-\beta _{1})^{2}\frac{\sum_{t=[\tau _{0}T]+1}^{[\tau_{0}T]+m}y_{t-1}^{2}\sum_{t=1}^{[\tau _{0}T]}y_{t-1}^{2}}{\sum_{t=1}^{[\tau
_{0}T]+m}y_{t-1}^{2}}+\Lambda _{T}(\tau _{0}+\frac{m}{T})
\end{eqnarray*}
where
\begin{eqnarray*}
&&\Lambda _{T}(\tau _{0}+\frac{m}{T}) \\
&=&\frac{\Big(\sum_{t=1}^{[\tau _{0}T]}y_{t-1}\varepsilon _{t}\Big)^{2}}{\sum_{t=1}^{[\tau _{0}T]}y_{t-1}^{2}}-\frac{\Big(\sum_{t=1}^{[\tau
_{0}T]+m}y_{t-1}\varepsilon _{t}\Big)^{2}}{\sum_{t=1}^{[\tau_{0}T]+m}y_{t-1}^{2}}+\frac{\Big(\sum_{t=[\tau
_{0}T]+1}^{T}y_{t-1}\varepsilon _{t}\Big)^{2}}{\sum_{t=[\tau_{0}T]+1}^{T}y_{t-1}^{2}}-\frac{\Big(\sum_{t=[\tau
_{0}T]+m+1}^{T}y_{t-1}\varepsilon _{t}\Big)^{2}}{\sum_{t=[\tau_{0}T]+m+1}^{T}y_{t-1}^{2}}.
\end{eqnarray*}
Since $\Lambda _{T}(\tau _{0}+\frac{m}{T})=O_{p}(l(\eta _{T}))$ and
\begin{equation*}
\left\vert \sum_{t=[\tau _{0}T]+1}^{[\tau _{0}T]+m}y_{t-1}\varepsilon_{t}\right\vert \leq \sqrt{\sum_{t=[\tau _{0}T]+1}^{[\tau
_{0}T]+m}\varepsilon _{t}^{2}}\cdot \sqrt{\sum_{t=[\tau _{0}T]+1}^{[\tau_{0}T]+m}y_{t-1}^{2}}=O_{p}(\sqrt{T}l(\eta _{T})),
\end{equation*}
we have
\begin{eqnarray*}
&&\frac{k_{T}^{2}}{Tl(\eta _{T})}\Big(RSS_{T}(\tau _{0}+\frac{m}{T})-RSS_{T}(\tau _{0})\Big) \\
&=&\frac{k_{T}^{2}}{Tl(\eta _{T})}\cdot \Big(-\frac{2c}{k_{T}}\Big)\cdot \bigg(\frac{O_{p}(T^{2}l(\eta _{T}))O_{p}(\sqrt{T}l(\eta _{T}))}{O_{p}(T^{2}l(\eta _{T}))+O_{p}(Tl(\eta _{T}))}+\frac{O_{p}(Tl(\eta_{T}))O_{p}(Tl(\eta _{T}))}{O_{p}(T^{2}l(\eta _{T}))+O_{p}(Tl(\eta _{T}))}\bigg) \\
&&+\frac{k_{T}^{2}}{Tl(\eta _{T})}\cdot \frac{c^{2}}{k_{T}^{2}}\cdot \frac{O_{p}(T^{2}l(\eta _{T}))}{O_{p}(T^{2}l(\eta _{T}))+O_{p}(Tl(\eta _{T}))}\sum_{t=[\tau _{0}T]+1}^{[\tau _{0}T]+m}y_{t-1}^{2}+o_{p}(1) \\
&=&\frac{c^{2}(1+o_{p}(1))}{Tl(\eta _{T})}\sum_{t=[\tau _{0}T]+1}^{[\tau_{0}T]+m}y_{t-1}^{2}+o_{p}(1) \\
&\Rightarrow &c^{2}mW^{2}(\tau _{0})
\end{eqnarray*}
by Lemma B.1 and $k_{T}=o(\sqrt{T})$. $\hfill \Box $\newline


\noindent \textbf{Proof of Lemma B.6.} \  Note that we have proven that $|\hat{\tau}_{T}-\tau _{0}|=O_{p}(k_{T}^{2}/T^{2})$ when $\sqrt{T}=o(k_T)$ in part (a) of Theorem 1.2. Then, for $\hat{\tau}_{T}\leq \tau_{0}$, we have
\begin{eqnarray}\label{B.19}
\sum_{t=[\tau _{0}T-k_{T}^{2}/T]+1}^{[\tau _{0}T]}y_{t-1}^{2}
&=&\sum_{t=[\tau _{0}T-k_{T}^{2}/T]+1}^{[\tau _{0}T]}\Big(y_{[\tau_{0}T]}-\sum_{i=t}^{[\tau _{0}T]}\varepsilon _{i}\Big)^{2}  \notag\\
&=&\sum_{t=[\tau _{0}T-k_{T}^{2}/T]+1}^{[\tau _{0}T]}y_{[\tau_{0}T]}^{2}\cdot (1+o_{p}(1))  \notag \\
&=&O_{p}(k_{T}^{2}l(\eta _{T})),
\end{eqnarray}
implying
\begin{equation}\label{B.6}
\sum_{t=[\hat{\tau}_{T}T]+1}^{[\tau _{0}T]}y_{t-1}^{2}=O_{p}(k_{T}^{2}l(\eta_{T})).
\end{equation}%
In addition, squaring $y_{t}=y_{t-1}+\varepsilon _{t}$ and summing over $t\in\{[\tau _{0}T-k_{T}^{2}/T]+1,\cdots ,[\tau _{0}T]\}$, we have
\begin{eqnarray*}
&&\sum_{t=[\tau _{0}T-k_{T}^{2}/T]+1}^{[\tau _{0}T]}y_{t-1}\varepsilon_{t}\\
&=&\frac{1}{2}\left( y_{[\tau _{0}T]}^{2}-y_{[\tau_{0}T-k_{T}^{2}/T]}^{2}-\sum_{t=[\tau _{0}T-k_{T}^{2}/T]+1}^{[\tau_{0}T]}\varepsilon _{t}^{2}\right) \\
&=&\frac{1}{2}\left( y_{[\tau _{0}T]}+y_{[\tau _{0}T-k_{T}^{2}/T]}\right)\left( y_{[\tau _{0}T]}-y_{[\tau _{0}T-k_{T}^{2}/T]}\right) -\frac{1}{2}\sum_{t=[\tau _{0}T-k_{T}^{2}/T]+1}^{[\tau _{0}T]}\varepsilon _{t}^{2} \\
&=&O_{p}(\sqrt{Tl(\eta _{T})})O_{p}(\sqrt{k_{T}^{2}l(\eta _{T})/T})+O_{p}(k_{T}^{2}l(\eta _{T})/T) \\
&=&O_{p}(k_{T}l(\eta _{T})).
\end{eqnarray*}
As a result,
\begin{equation}
\sum_{t=[\hat{\tau}_{T}T]+1}^{[\tau _{0}T]}y_{t-1}\varepsilon_{t}=O_{p}(k_{T}l(\eta _{T})).  \label{B.7}
\end{equation}%
Similarly, if $\hat{\tau}_{T}>\tau _{0}$, it follows from Lemma B.2 that
\begin{eqnarray}\label{B.2}
\sum_{t=[\tau _{0}T]+1}^{[\tau _{0}T+k_{T}^{2}/T]}y_{t-1}^{2}
&=&\sum_{t=[\tau _{0}T]+1}^{[\tau _{0}T+k_{T}^{2}/T]}\Big(\beta_{2T}^{t-1-[\tau _{0}T]}y_{[\tau _{0}T]}+\sum_{i=[\tau _{0}T]+1}^{t-1}\beta_{2T}^{t-1-i}\varepsilon _{i}\Big)^{2}  \notag  \\
&=&\sum_{t=[\tau _{0}T]+1}^{[\tau _{0}T+k_{T}^{2}/T]}\beta_{2T}^{2(t-1-[\tau _{0}T])}y_{[\tau _{0}T]}^{2}\cdot (1+o_{p}(1))  \notag \\
&=&k_{T}^{2}/T\cdot O_{p}(Tl(\eta_{T}))  \notag \\
&=&O_{p}(k_{T}^{2}l(\eta _{T})),
\end{eqnarray}
implying
\begin{equation}\label{B.13}
\sum_{t=[\tau _{0}T]+1}^{[\hat{\tau}_{T}T]}y_{t-1}^{2}=O_{p}(k_{T}^{2}l(\eta_{T})).
\end{equation}
By squaring $y_{t}=\beta _{2T}y_{t-1}+\varepsilon _{t}$ and summing over $t\in \{[\tau _{0}T]+1,\cdots ,[\tau _{0}T+k_{T}^{2}/T]\}$, we obtain
\begin{equation}
\sum_{t=[\tau _{0}T]+1}^{[\tau_{0}T+k_{T}^{2}/T]}y_{t-1}\varepsilon_{t}=\frac{1-\beta _{2T}^{2}}{2\beta _{2T}}\sum_{t=[\tau _{0}T]+1}^{[\tau
_{0}T+k_{T}^{2}/T]}y_{t-1}^{2}+\frac{y_{[\tau_{0}T+k_{T}^{2}/T]}^{2}-y_{[\tau _{0}T]}^{2}}{2\beta _{2T}}-\frac{1}{2\beta_{2T}}\sum_{t=[\tau _{0}T]+1}^{[\tau _{0}T+k_{T}^{2}/T]}\varepsilon_{t}^{2}.
\label{B.14}
\end{equation}
First, applying (\ref{B.2}) immediately yields
\begin{equation}\label{B.15}
\frac{1-\beta_{2T}^{2}}{2\beta_{2T}}\sum_{t=[\tau _{0}T]+1}^{[\tau_{0}T+k_{T}^{2}/T]}y_{t-1}^{2}=O_{p}(k_{T}l(\eta _{T})).
\end{equation}
Secondly, since $\sqrt{T}=o(k_{T})$, we have
\begin{equation}\label{B.16}
\frac{1}{2\beta _{2T}}\sum_{t=[\tau _{0}T]+1}^{[\tau_{0}T+k_{T}^{2}/T]}\varepsilon _{t}^{2}=O_{p}(k_{T}^{2}l(\eta _{T})/T).
\end{equation}%
Thirdly, since $\left\vert \beta _{2T}\right\vert <1$, we have
\begin{equation*}
\sum_{i=0}^{[k_{T}^{2}/T]-1}\beta _{2T}^{2i}=O(k_{T}^{2}/T)
\end{equation*}
which implies that
\begin{equation*}
\sum_{i=0}^{[k_{T}^{2}/T]-1}\beta _{2T}^{i}\varepsilon _{\lbrack \tau_{0}T+k_{T}^{2}/T]-i}=O_{p}(k_{T}\sqrt{l(\eta _{T})}/\sqrt{T}).
\end{equation*}%
Then, we have
\begin{eqnarray}\label{B.17}
&&y_{[\tau_{0}T+k_{T}^{2}/T]}^{2}-y_{[\tau _{0}T]}^{2}  \notag\\
&=&\left( y_{[\tau _{0}T+k_{T}^{2}/T]}+y_{[\tau _{0}T]}\right) \left(y_{[\tau _{0}T+k_{T}^{2}/T]}-y_{[\tau _{0}T]}\right)  \notag \\
&=&\left( \sum_{i=0}^{[k_{T}^{2}/T]-1}\beta _{2T}^{i}\varepsilon _{\lbrack\tau _{0}T+k_{T}^{2}/T]-i}+(\beta _{2T}^{[k_{T}^{2}/T]}+1)y_{[\tau
_{0}T]}\right)  \notag \\
&&\cdot \left( \sum_{i=0}^{[k_{T}^{2}/T]-1}\beta _{2T}^{i}\varepsilon_{\lbrack \tau _{0}T+k_{T}^{2}/T]-i}+(\beta _{2T}^{[k_{T}^{2}/T]}-1)y_{[\tau
_{0}T]}\right)  \notag \\
&=&\left( O_{p}(k_{T}\sqrt{l(\eta _{T})}/\sqrt{T})+O_{p}(\sqrt{Tl(\eta _{T})})\right) \left( O_{p}(k_{T}\sqrt{l(\eta _{T})}/\sqrt{T})+\frac{k_{T}^{2}/T}{k_{T}}O_{p}(\sqrt{Tl(\eta _{T})}\right)  \notag \\
&=&O_{p}(\sqrt{Tl(\eta _{T})})O_{p}(k_{T}\sqrt{l(\eta _{T})}/\sqrt{T})\notag \\
&=&O_{p}(k_{T}l(\eta _{T})).
\end{eqnarray}%
Combining (\ref{B.14})-(\ref{B.17}) together leads to
\begin{equation}\label{B.1}
\sum_{t=[\tau_{0}T]+1}^{[\tau _{0}T+k_{T}^{2}/T]}y_{t-1}\varepsilon_{t}=O_{p}(k_{T}l(\eta _{T})),
\end{equation}
which implies
\begin{equation}\label{B.18}
\sum_{t=[\tau_{0}T]+1}^{[\hat{\tau}_{T}T]}y_{t-1}\varepsilon_{t}=O_{p}(k_{T}l(\eta _{T})).
\end{equation}
$\hfill \Box $\newline



\noindent\textbf{Proof of Lemma C.1.} \; The proof can be completed by following the proof of Lemma 4.2 in Phillips and Magdalinos (2007a) and the truncation
technique (A.1). Thus the details are omitted. $\hfill \Box $\newline


\noindent\textbf{Proof of Lemma C.2.} \; To prove part (a), we first note that the following decomposition holds:
\begin{eqnarray}  \label{C.23}
&&\frac{1}{\beta_{2T}^{T-[\tau_0 T]}\sqrt{T k_T} l(\eta_T)}\sum_{t=[\tau_0 T]+1}^{T}y_{t-1}\varepsilon_t  \notag \\
&=&\frac{1}{\beta_{2T}^{T-[\tau_0 T]}\sqrt{T k_T} l(\eta_T)}\sum_{t=[\tau_0 T] +1}^{T}\Big(\beta_{2T}^{t-[\tau_0 T] -1}y_{[\tau_0 T] }+\sum_{i=[\tau_0
T] +1}^{t-1}\beta_{2T}^{t-i-1}\varepsilon_i\Big)\varepsilon_t  \notag \\
&=&\frac{1}{\beta_{2T}^{T-[\tau_0 T]}\sqrt{T k_T} l(\eta_T)}\left(\Big(y_0+\sum_{j=1}^{[\tau_0 T] }\varepsilon_j\Big)\sum_{t=[\tau_0 T]
+1}^{T}\beta_{2T}^{t-[\tau_0 T] -1}\varepsilon_t+ \sum_{t=[\tau_0 T]+1}^{T}\varepsilon_t\sum_{i=[\tau_0 T]
+1}^{t-1}\beta_{2T}^{t-i-1}\varepsilon_i\right)  \notag \\
&:=&II_1+II_2,
\end{eqnarray}
where
\begin{eqnarray*}
II_1&=&\frac{1}{\beta_{2T}^{T-[\tau_0 T]}\sqrt{T k_T} l(\eta_T)}\Big(y_0+\sum_{j=1}^{[\tau_0 T] }\varepsilon_j\Big)\sum_{t=[\tau_0 T]
+1}^{T}\beta_{2T}^{t-[\tau_0 T] -1}\varepsilon_t  \notag \\
&=&\frac{1}{\beta_{2T}^{T-[\tau_0 T]}\sqrt{T k_T} l(\eta_T)}\sum_{j=1}^{[\tau_0 T] }\varepsilon_j\sum_{t=[\tau_0 T]+1}^{T}\beta_{2T}^{t-[\tau_0 T] -1}\varepsilon_t\cdot (1+o_p(1))
\end{eqnarray*}
and
\begin{eqnarray*}
II_2&=&\frac{1}{\beta_{2T}^{T-[\tau_0 T]}\sqrt{T k_T} l(\eta_T)}\sum_{t=[\tau_0 T] +1}^{T}\varepsilon_t\sum_{i=[\tau_0 T]
+1}^{t-1}\beta_{2T}^{t-i-1}\varepsilon_i  \notag \\
&=&\frac{1}{\beta_{2T}^{T-[\tau_0 T]}\sqrt{T k_T} l(\eta_T)}\sum_{t=[\tau_0 T] +1}^{T}\varepsilon_t^{(1)}\sum_{i=[\tau_0 T]
+1}^{t-1}\beta_{2T}^{t-i-1}\varepsilon_i^{(1)}\cdot (1+o_p(1))
\end{eqnarray*}
by the truncation technique (A.1) and some simple calculus.

For the term $II_{1}$, applying part (a) of Lemma C.1 and observing the fact $\frac{\sum_{t=1}^{[\tau _{0}T]}\varepsilon _{t}}{\sqrt{Tl(\eta _{T})}}\Rightarrow W(\tau _{0})$, it can be shown that
\begin{equation}  \label{C.21}
II_{1}\Rightarrow XW(\tau _{0})
\end{equation}
by noting that
\begin{eqnarray*}
&&\frac{1}{\beta _{2T}^{T-[\tau _{0}T]}\sqrt{k_{T}l(\eta _{T})}}\sum_{t=[\tau _{0}T]+1}^{T}\beta _{2T}^{t-[\tau _{0}T]-1}\varepsilon _{t} \\
&=&\frac{1}{\beta _{2T}^{T-[\tau _{0}T]}\sqrt{k_{T}l(\eta _{T})}}\sum_{s=1}^{T-[\tau _{0}T]}\beta _{2T}^{s-1}\varepsilon _{\lbrack \tau_{0}T]+s} \\
&=&\frac{1}{\sqrt{k_{T}l(\eta _{T})}}\sum_{s=1}^{[(1-\tau _{0})T]}\beta_{2T}^{s-1-[(1-\tau _{0})T]}\varepsilon _{\lbrack \tau _{0}T]+s}\cdot(1+o_{p}(1))
\end{eqnarray*}
and
\begin{equation*}
\frac{1}{\sqrt{k_{T}l(\eta _{T})}}\sum_{s=1}^{[(1-\tau _{0})T]}\beta_{2T}^{s-1-[(1-\tau _{0})T]}\varepsilon _{\lbrack \tau _{0}T]+s}\overset{d}{=} \frac{1}{\sqrt{k_{T}l(\eta _{T})}}\sum_{s=1}^{[(1-\tau _{0})T]}\beta_{2T}^{s-1-[(1-\tau _{0})T]}\varepsilon _{s}\Rightarrow X.
\end{equation*}

For the term $II_2$, it is trivial that
\begin{eqnarray*}
E\Big(\frac{1}{\beta_{2T}^{T-[\tau_0 T]}\sqrt{T k_T} l(\eta_T)}\sum_{t=[\tau_0 T] +1}^{T}\varepsilon_t^{(1)}\sum_{i=[\tau_0 T]+1}^{t-1}\beta_{2T}^{t-i-1}\varepsilon_i^{(1)}\Big)=0
\end{eqnarray*}
and
\begin{eqnarray}  \label{C.14}
&&Var\Big(\frac{1}{\beta_{2T}^{T-[\tau_0 T]}\sqrt{T k_T} l(\eta_T)}\sum_{t=[\tau_0 T] +1}^{T}\varepsilon_t^{(1)}\sum_{i=[\tau_0 T]
+1}^{t-1}\beta_{2T}^{t-i-1}\varepsilon_i^{(1)}\Big)  \notag \\
&=&\frac{1}{\beta_{2T}^{2(T-[\tau_0 T])}T k_T}\sum_{t=[\tau_0 T]+1}^{T}\sum_{i=[\tau_0 T] +1}^{t-1}\beta_{2T}^{2(t-i-1)}\cdot (1+o(1))
\notag \\
&=&\frac{1}{\beta_{2T}^{2(T-[\tau_0 T])}T }\cdot \frac{1}{k_T(\beta_{2T}^2-1)}\cdot \Big(\frac{\beta_{2T}^{2(T-[\tau_0 T])}-1}{\beta_{2T}^2-1}-(T-[\tau_0 T])\Big)\cdot (1+o(1))  \notag \\
&=&o(1)
\end{eqnarray}
by $k_T=o(T)$. Consequently,
\begin{eqnarray}  \label{C.22}
II_2=o_p(1).
\end{eqnarray}
Obviously, combining (\ref{C.23}), (\ref{C.21}) and (\ref{C.22}) yields part (a).

To prove part (b), we write
\begin{eqnarray}  \label{C.18}
&&\frac{1}{\beta_{2T}^{2(T-[\tau_0 T])}T k_T l(\eta_T)}\sum_{t=[\tau_0 T]+1}^{T}y_{t-1}^2  \notag \\
&=&\frac{1}{\beta_{2T}^{2(T-[\tau_0 T])}T k_T l(\eta_T)}\sum_{t=[\tau_0 T]+1}^{T}\Big(\beta_{2T}^{t-[\tau_0 T] -1}y_{[\tau_0 T] }+\sum_{i=[\tau_0 T]
+1}^{t-1}\beta_{2T}^{t-i-1}\varepsilon_i\Big)^2  \notag \\
&=&\frac{y_{[\tau_0 T]}^2}{\beta_{2T}^{2(T-[\tau_0 T])}T k_T l(\eta_T)}\sum_{t=[\tau_0 T]+1}^{T}\beta_{2T}^{2(t-[\tau_0 T]-1)}+\frac{1}{%
\beta_{2T}^{2(T-[\tau_0 T])}T k_T l(\eta_T)}\sum_{t=[\tau_0 T]+1}^{T}\Big(\sum_{i=[\tau_0 T] +1}^{t-1}\beta_{2T}^{t-i-1}\varepsilon_i\Big)^2  \notag \\
&&+\frac{2 y_{[\tau_0 T]}}{\beta_{2T}^{2(T-[\tau_0 T])}T k_T l(\eta_T)}\sum_{t=[\tau_0 T]+1}^{T}\Big(\beta_{2T}^{t-[\tau_0 T] -1}\sum_{i=[\tau_0 T]
+1}^{t-1}\beta_{2T}^{t-i-1}\varepsilon_i\Big)  \notag \\
&:=&III_1+III_2+III_3,
\end{eqnarray}
where
\begin{eqnarray*}
III_1=\frac{y_{[\tau_0 T]}^2}{\beta_{2T}^{2(T-[\tau_0 T])}T k_T l(\eta_T)}\sum_{t=[\tau_0 T]+1}^{T}\beta_{2T}^{2(t-[\tau_0 T]-1)},
\end{eqnarray*}
\begin{eqnarray*}
III_2=\frac{1}{\beta_{2T}^{2(T-[\tau_0 T])}T k_T l(\eta_T)}\sum_{t=[\tau_0 T]+1}^{T}\Big(\sum_{i=[\tau_0 T] +1}^{t-1}\beta_{2T}^{t-i-1}\varepsilon_i%
\Big)^2
\end{eqnarray*}
and
\begin{eqnarray*}
III_3=\frac{2 y_{[\tau_0 T]}}{\beta_{2T}^{2(T-[\tau_0 T])}T k_T l(\eta_T)}\sum_{t=[\tau_0 T]+1}^{T}\Big(\beta_{2T}^{t-[\tau_0 T] -1}\sum_{i=[\tau_0 T]
+1}^{t-1}\beta_{2T}^{t-i-1}\varepsilon_i\Big).
\end{eqnarray*}

Obviously,
\begin{equation}  \label{C.15}
III_{1}\Rightarrow \frac{1}{2c}W^{2}(\tau _{0}).
\end{equation}

For the term $III_{2}$, it is true that
\begin{equation*}
III_{2}=\frac{1}{\beta _{2T}^{2(T-[\tau _{0}T])}Tk_{T}l(\eta _{T})}\sum_{t=[\tau _{0}T]+1}^{T}\Big(\sum_{i=[\tau _{0}T]+1}^{t-1}\beta_{2T}^{t-i-1}\varepsilon _{i}^{(1)}\Big)^{2}\cdot (1+o_{p}(1))
\end{equation*}
and
\begin{eqnarray*}
&&E\left( \frac{1}{\beta _{2T}^{2(T-[\tau _{0}T])}Tk_{T}l(\eta _{T})}\sum_{t=[\tau _{0}T]+1}^{T}\Big(\sum_{i=[\tau _{0}T]+1}^{t-1}\beta
_{2T}^{t-i-1}\varepsilon _{i}^{(1)}\Big)^{2}\right) \\
&=&\frac{1}{\beta _{2T}^{2(T-[\tau _{0}T])}Tk_{T}}\sum_{t=[\tau_{0}T]+1}^{T}\sum_{i=[\tau _{0}T]+1}^{t-1}\beta _{2T}^{2(t-i-1)}\cdot
(1+o(1)) \\
&=&o(1)
\end{eqnarray*}%
by similar arguments as in (\ref{C.14}). Hence, we have
\begin{equation}  \label{C.16}
III_{2}=o_{p}(1).
\end{equation}

For the term $III_{3}$, using (\ref{C.15}) and (\ref{C.16}) and applying Cauchy-Schwarz inequality immediately leads to
\begin{equation}  \label{C.17}
III_{3}=o_{p}(1).
\end{equation}

Combining (\ref{C.18})-(\ref{C.17}), we show part (b).

Moreover, by checking the above proofs carefully, one can find that parts (a) and (b) hold jointly and $X$ and $W(\tau _{0})$ are mutually independent.  $\hfill \Box $\newline


\noindent \textbf{Proof of Lemma C.3.} \ First, it
is trivial that
\begin{equation}\label{C.2}
A_{1}=o_{p}(1/k_{T}),\quad A_{4}=o_{p}(1/k_{T})
\end{equation}
by Lemma B.1 and Lemma C.2. Secondly, note that the same reasoning in (\ref{B.4}) also implies
\begin{equation}  \label{C.3}
A_{2}=o_{p}(1/k_{T}).
\end{equation}
For the term $A_{5}$, following the proof of Theorem 1.2, we have
\begin{eqnarray*}
&&y_{[\tau_{0}T]}\sum_{t=[\tau _{0}T]+1}^{m}\beta _{2T}^{t-1-[\tau_{0}T]}\varepsilon _{t} \\
&=&O_{p}\left( \sqrt{Tk_{T}(\beta _{2T}^{2(m-[\tau _{0}T])}-1)}l(\eta_{T})\right) \\
&=&\left\{
\begin{array}{ll}
O_{p}\left( \sqrt{T(m-[\tau _{0}T])}l(\eta _{T})\right) ,~ & \mbox{if}~M_{T}<m-[\tau _{0}T]\leq \lbrack k_{T}/N_{1}] \\
O_{p}\left( \sqrt{Tk_{T}}l(\eta _{T})\right) ,~ & \mbox{if}~[k_{T}/N_{1}]<m-[\tau _{0}T]\leq \lbrack N_{2}k_{T}] \\
O_{p}\left( \sqrt{Tk_{T}}\beta _{2T}^{(m-[\tau _{0}T])}l(\eta _{T})\right),~& \mbox{if}~[N_{2}k_{T}]<m-[\tau _{0}T]\leq T-[\tau _{0}T]%
\end{array}
\right.
\end{eqnarray*}
and
\begin{eqnarray*}
&&\sum_{t=[\tau _{0}T]+1}^{m}\left( \sum_{i=[\tau _{0}T]+1}^{t-1}\beta_{2T}^{t-1-i}\varepsilon _{i}\right) \varepsilon _{t} \\
&=&O_{p}\left( \sqrt{k_{T}(m-[\tau _{0}T])+k_{T}^{2}(\beta _{2T}^{2(m-[\tau_{0}T])}-1)}l(\eta _{T})\right) \\
&=&\left\{
\begin{array}{ll}
O_{p}\left( \sqrt{k_{T}(m-[\tau _{0}T])}l(\eta _{T})\right) ,~ & \mbox{if}~M_{T}<m-[\tau _{0}T]\leq \lbrack k_{T}/N_{1}] \\
O_{p}\left( k_{T}l(\eta _{T})\right) ,~ & \mbox{if}~[k_{T}/N_{1}]<m-[\tau_{0}T]\leq \lbrack N_{2}k_{T}] \\
O_{p}\left( k_{T}\beta _{2T}^{m-[\tau _{0}T]}l(\eta _{T})\right) ,~ &\mbox{if}~[N_{2}k_{T}]<m-[\tau _{0}T]\leq T-[\tau _{0}T]%
\end{array}.\right.
\end{eqnarray*}
Hence,
\begin{eqnarray}\label{C.30}
&&\sum_{t=[\tau _{0}T]+1}^{m}y_{t-1}\varepsilon_{t}  \notag  \\
&=&y_{[\tau_{0}T]}\sum_{t=[\tau_{0}T]+1}^{m}\beta_{2T}^{t-1-[\tau_{0}T]}\varepsilon _{t}+\sum_{t=[\tau _{0}T]+1}^{m}\left( \sum_{i=[\tau_{0}T]+1}^{t-1}\beta _{2T}^{t-1-i}\varepsilon _{i}\right) \varepsilon_{t}
\notag \\
&=&\left\{
\begin{array}{ll}
O_{p}\left( \sqrt{T(m-[\tau _{0}T])}l(\eta _{T})\right) ,~ & \mbox{if}~M_{T}<m-[\tau _{0}T]\leq \lbrack k_{T}/N_{1}] \\
O_{p}\left( \sqrt{Tk_{T}}l(\eta _{T})\right) ,~ & \mbox{if}~[k_{T}/N_{1}]<m-[\tau _{0}T]\leq \lbrack N_{2}k_{T}] \\
O_{p}\left( \sqrt{Tk_{T}}\beta _{2T}^{(m-[\tau _{0}T])}l(\eta _{T})\right),~& \mbox{if}~[N_{2}k_{T}]<m-[\tau _{0}T]\leq T-[\tau _{0}T]%
\end{array}.\right.
\end{eqnarray}
Moreover, we also have
\begin{eqnarray*}
&&\frac{2\beta _{2T}}{1-\beta _{2T}^{2}}\sum_{t=[\tau_{0}T]+1}^{m}y_{t-1}\varepsilon _{t} \\
&=&\left\{
\begin{array}{ll}
O_{p}\left( k_{T}\sqrt{T(m-[\tau _{0}T])}l(\eta _{T})\right) ,~ & \mbox{if}~M_{T}<m-[\tau _{0}T]\leq \lbrack k_{T}/N_{1}] \\
O_{p}\left( \sqrt{T}k_{T}^{3/2}l(\eta _{T})\right) ,~ & \mbox{if}~[k_{T}/N_{1}]<m-[\tau _{0}T]\leq \lbrack N_{2}k_{T}] \\
O_{p}\left( \sqrt{T}k_{T}^{3/2}\beta _{2T}^{(m-[\tau _{0}T])}l(\eta_{T})\right),~ & \mbox{if}~[N_{2}k_{T}]<m-[\tau _{0}T]\leq T-[\tau _{0}T]
\end{array},\right.
\end{eqnarray*}
\begin{eqnarray*}
&&\frac{1}{1-\beta_{2T}^{2}}\sum_{t=[\tau_{0}T]+1}^{m}\varepsilon_{t}^{2}\\
&=&O_{p}\left( k_{T}(m-[\tau _{0}T])l(\eta _{T})\right) \\
&=&\left\{
\begin{array}{ll}
O_{p}\left( k_{T}(m-[\tau _{0}T])l(\eta _{T})\right) ,~ & \mbox{if}~M_{T}<m-[\tau _{0}T]\leq \lbrack k_{T}/N_{1}] \\
O_{p}\left( k_{T}^{2}l(\eta _{T})\right) ,~ & \mbox{if}~[k_{T}/N_{1}]<m-[\tau _{0}T]\leq \lbrack N_{2}k_{T}] \\
O_{p}\left( k_{T}(m-[\tau _{0}T])l(\eta _{T})\right) ,~ & \mbox{if}~[N_{2}k_{T}]<m-[\tau _{0}T]\leq T-[\tau _{0}T]%
\end{array}
\right.
\end{eqnarray*}
and
\begin{eqnarray*}
&&-\frac{1}{1-\beta _{2T}^{2}}(y_{m}^{2}-y_{[\tau _{0}T]}^{2}) \\
&=&-\frac{1}{1-\beta _{2T}^{2}}(y_{m}+y_{[\tau _{0}T]})(y_{m}-y_{[\tau_{0}T]}) \\
&=&-\frac{1}{1-\beta _{2T}^{2}}\left( \sum_{i=[\tau _{0}T]+1}^{m}\beta_{2T}^{m-i}\varepsilon _{i}+\left( \beta _{2T}^{m-[\tau _{0}T]}+1\right)
y_{[\tau _{0}T]}\right) \\
&&\cdot \left( \sum_{i=[\tau _{0}T]+1}^{m}\beta _{2T}^{m-i}\varepsilon_{i}+\left( \beta _{2T}^{m-[\tau _{0}T]}-1\right) y_{[\tau _{0}T]}\right) \\
&=&k_{T}\cdot \left( O_{p}\left( \sqrt{k_{T}(\beta _{2T}^{2(m-[\tau_{0}T])}-1)l(\eta _{T})}\right) +O_{p}(\beta _{2T}^{m-[\tau _{0}T]}\sqrt{Tl(\eta _{T})})\right) \\
&&\cdot \left( O_{p}\left( \sqrt{k_{T}(\beta _{2T}^{2(m-[\tau_{0}T])}-1)l(\eta _{T})}\right) +O_{p}\left( (\beta _{2T}^{m-[\tau _{0}T]}-1)\sqrt{Tl(\eta _{T})}\right) \right) \\
&=&\left\{
\begin{array}{ll}
O_{p}(k_{T}\sqrt{Tl(\eta _{T})})\cdot O_{p}\left( \frac{m-[\tau _{0}T]}{k_{T}}\sqrt{Tl(\eta _{T})}\right) ,~ & \mbox{if}~M_{T}<m-[\tau _{0}T]\leq \lbrack
k_{T}/N_{1}] \\
O_{p}(k_{T}\sqrt{Tl(\eta _{T})})\cdot O_{p}(\sqrt{Tl(\eta _{T})}),~ & \mbox{if}~[k_{T}/N_{1}]<m-[\tau _{0}T]\leq \lbrack N_{2}k_{T}] \\
O_{p}(k_{T}\beta _{2T}^{m-[\tau _{0}T]}\sqrt{Tl(\eta _{T})})\cdot
O_{p}(\beta _{2T}^{m-[\tau _{0}T]}\sqrt{Tl(\eta _{T})}),~ & \mbox{if}~[N_{2}k_{T}]<m-[\tau _{0}T]\leq T-[\tau _{0}T]%
\end{array}
\right. \\
&=&\left\{
\begin{array}{ll}
O_{p}\left( T(m-[\tau _{0}T])l(\eta _{T})\right) ,~ & \mbox{if}~M_{T}<m-[\tau _{0}T]\leq \lbrack k_{T}/N_{1}] \\
O_{p}(Tk_{T}l(\eta _{T})),~ & \mbox{if}~[k_{T}/N_{1}]<m-[\tau _{0}T]\leq\lbrack N_{2}k_{T}] \\
O_{p}(Tk_{T}\beta _{2T}^{2(m-[\tau _{0}T])}l(\eta _{T})),~ & \mbox{if}~[N_{2}k_{T}]<m-[\tau _{0}T]\leq T-[\tau _{0}T]%
\end{array}.\right.
\end{eqnarray*}
Thus,
\begin{eqnarray}\label{C.31}
&&\sum_{t=[\tau_{0}T]+1}^{m}y_{t-1}^{2}  \notag  \\
&=&-\frac{1}{1-\beta _{2T}^{2}}(y_{m}^{2}-y_{[\tau _{0}T]}^{2})+\frac{1}{1-\beta _{2T}^{2}}\sum_{t=[\tau _{0}T]+1}^{m}\varepsilon _{t}^{2}+\frac{2\beta _{2T}}{1-\beta _{2T}^{2}}\sum_{t=[\tau_{0}T]+1}^{m}y_{t-1}\varepsilon _{t}  \notag \\
&=&\left\{
\begin{array}{ll}
O_{p}\left( T(m-[\tau _{0}T])l(\eta _{T})\right) ,~ & \mbox{if}~M_{T}<m-[\tau _{0}T]\leq \lbrack k_{T}/N_{1}] \\
O_{p}(Tk_{T}l(\eta _{T})),~ & \mbox{if}~[k_{T}/N_{1}]<m-[\tau _{0}T]\leq\lbrack N_{2}k_{T}] \\
O_{p}(Tk_{T}\beta _{2T}^{2(m-[\tau _{0}T])}l(\eta _{T})),~ & \mbox{if}~[N_{2}k_{T}]<m-[\tau _{0}T]\leq T-[\tau _{0}T]%
\end{array}.\right.
\end{eqnarray}
Consequently, it follows from (\ref{C.30}) and (\ref{C.31}) that
\begin{equation*}
\frac{\sum_{t=[\tau _{0}T]+1}^{m}y_{t-1}\varepsilon _{t}}{\sum_{t=[\tau_{0}T]+1}^{m}y_{t-1}^{2}}=O_{p}\left( \frac{\sqrt{T(m-[\tau _{0}T])}l(\eta
_{T})}{T(m-[\tau _{0}T])l(\eta _{T})}\right) \leq O_{p}(1/\sqrt{TM_{T}})=o_{p}(1/k_{T})
\end{equation*}
by $k_{T}=O(\sqrt{T})$ and $M_{T}\rightarrow \infty $ when $M_{T}<m-[\tau_{0}T]\leq \lbrack k_{T}/N_{1}]$,
\begin{equation*}
\frac{\sum_{t=[\tau_{0}T]+1}^{m}y_{t-1}\varepsilon _{t}}{\sum_{t=[\tau_{0}T]+1}^{m}y_{t-1}^{2}}=O_{p}\left(\frac{\sqrt{Tk_{T}}l(\eta_{T})}{Tk_{T}l(\eta _{T})}\right) =o_{p}(1/k_{T})
\end{equation*}
when $[k_{T}/N_{1}]<m-[\tau _{0}T]\leq \lbrack N_{2}k_{T}]$ and
\begin{equation*}
\frac{\sum_{t=[\tau_{0}T]+1}^{m}y_{t-1}\varepsilon_{t}}{\sum_{t=[\tau_{0}T]+1}^{m}y_{t-1}^{2}}=O_{p}\left( \frac{\sqrt{Tk_{T}}\beta
_{2T}^{m-[\tau _{0}T]}l(\eta _{T})}{Tk_{T}\beta _{2T}^{2(m-[\tau_{0}T])}l(\eta _{T})}\right) =o_{p}(1/k_{T})
\end{equation*}
when $[N_{2}k_{T}]<m-[\tau _{0}T]\leq T-[\tau _{0}T]$. These imply
\begin{equation}\label{C.32}
A_{5}=\sup_{m\in D_{2T}}\frac{\sum_{t=[\tau _{0}T]+1}^{m}y_{t-1}\varepsilon_{t}}{\sum_{t=[\tau _{0}T]+1}^{m}y_{t-1}^{2}}=o_{p}(1/k_{T}).
\end{equation}
Thirdly, it is clear that
\begin{equation}\label{C.35}
A_{3}\leq \Big(O_{p}\big(\frac{1}{Tk_{T}\beta _{2T}^{2(T-[\tau_{0}T])}l(\eta _{T})}\big)+O_{p}\big(\frac{1}{TM_{T}l(\eta _{T})}\big)\Big)\cdot O_{p}(l(\eta _{T}))=o_{p}(1/k_{T}^{2})
\end{equation}
and
\begin{equation}\label{C.6}
A_{6}\leq \Big(O_{p}(\frac{1}{T^{2}l(\eta _{T})})+O_{p}(\frac{1}{TM_{T}l(\eta _{T})})\Big)\cdot O_{p}(l(\eta _{T}))=o_{p}(1/k_{T}^{2}).
\end{equation}
The proofs are complete. $\hfill \Box $\newline


\noindent \textbf{Proof of Lemma C.4.} \ Note that $|\hat{\tau}_{T}-\tau _{0}|=O_{p}(k_{T}^{2}/T^{2})$ when $\sqrt{T}=o(k_T)$. Then, for $\hat{\tau}_{T}\leq \tau_{0}$, we have
\begin{equation}\label{C.7}
\sum_{t=[\hat{\tau}_{T}T]+1}^{[\tau _{0}T]}y_{t-1}^{2}=O_{p}(k_{T}^{2}l(\eta_{T}))
\end{equation}
and
\begin{equation}\label{C.8}
\sum_{t=[\hat{\tau}_{T}T]+1}^{[\tau _{0}T]}y_{t-1}\varepsilon_{t}=O_{p}(k_{T}l(\eta _{T}))
\end{equation}
by (\ref{B.6}) and (\ref{B.7}). For $\hat{\tau}_{T}>\tau _{0}$, it follows from the similar arguments in the proofs of (\ref{B.2}) and (\ref{B.1})
respectively that
\begin{eqnarray*}
\sum_{t=[\tau _{0}T]+1}^{[\tau _{0}T+k_{T}^{2}/T]}y_{t-1}^{2}&=&\sum_{t=[\tau _{0}T]+1}^{[\tau _{0}T+k_{T}^{2}/T]}\Big(\beta_{2T}^{t-1-[\tau _{0}T]}y_{[\tau _{0}T]}+\sum_{i=[\tau _{0}T]+1}^{t-1}\beta_{2T}^{t-1-i}\varepsilon _{i}\Big)^{2} \\
&=&\sum_{t=[\tau _{0}T]+1}^{[\tau _{0}T+k_{T}^{2}/T]}\Big(\beta_{2T}^{t-1-[\tau _{0}T]}y_{[\tau _{0}T]}+O_{p}(\sqrt{k_{T}\beta_{2T}^{t-1-[\tau _{0}T]}l(\eta _{T})})\Big)^{2} \\
&=&\sum_{t=[\tau _{0}T]+1}^{[\tau _{0}T+k_{T}^{2}/T]}\beta_{2T}^{2(t-1-[\tau _{0}T])}y_{[\tau _{0}T]}^{2}\cdot (1+o_{p}(1)) \\
&=&k_{T}^{2}/T\cdot O_{p}(Tl(\eta _{T})) \\
&=&O_{p}(k_{T}^{2}l(\eta _{T})),
\end{eqnarray*}
yielding
\begin{equation}\label{C.9}
\sum_{t=[\tau _{0}T]+1}^{[\hat{\tau}_{T}T]}y_{t-1}^{2}=O_{p}(k_{T}^{2}l(\eta_{T}))
\end{equation}
and
\begin{eqnarray*}
\sum_{t=[\tau _{0}T]+1}^{[\tau _{0}T+k_{T}^{2}/T]}y_{t-1}\varepsilon _{t} &=&\frac{1-\beta _{2T}^{2}}{2\beta _{2T}}\sum_{t=[\tau _{0}T]+1}^{[\tau
_{0}T+k_{T}^{2}/T]}y_{t-1}^{2}+\frac{y_{[\tau_{0}T+k_{T}^{2}/T]}^{2}-y_{[\tau _{0}T]}^{2}}{2\beta _{2T}}-\frac{1}{2\beta_{2T}}\sum_{t=[\tau _{0}T]+1}^{[\tau _{0}T+k_{T}^{2}/T]}\varepsilon _{t}^{2}\\
&=&O_{p}(k_{T}l(\eta _{T}))+O_{p}(\sqrt{Tl(\eta _{T})})\cdot O_{p}(k_{T}\sqrt{l(\eta _{T})}/\sqrt{T})+O_{p}(k_{T}^{2}l(\eta _{T})/T) \\
&=&O_{p}(k_{T}l(\eta _{T})),
\end{eqnarray*}
yielding
\begin{equation} \label{C.10}
\sum_{t=[\tau _{0}T]+1}^{[\hat{\tau}_{T}T]}y_{t-1}\varepsilon_{t}=O_{p}(k_{T}l(\eta _{T})).
\end{equation}
$\hfill \Box $\newline


\noindent \textbf{Proof of Lemma C.5.} \ The proof is similar to that of Lemma B.5, with the use of the result in (C.1). The details are
omitted. $\hfill \Box $\newline



\noindent \textbf{Proof of Lemma D.1.} \ The results can be proved by following the proof of Theorem 4.3 in Phillips and Magdalinos (2007a) and applying the truncation technique as shown in (A.1). The details are omitted. $\hfill \Box $\newline


\noindent\textbf{Proof of Lemma D.2.} \; To prove part (a), we write
\begin{eqnarray}  \label{D.0}
&&\frac{\beta_{1T}^{-[\tau_0 T]}}{\sqrt{T k_T} l(\eta_T)}\sum_{t=[\tau_0 T]+1}^{T}y_{t-1}\varepsilon_t  \notag \\
&=&\frac{\beta_{1T}^{-[\tau_0 T]}}{\sqrt{T k_T} l(\eta_T)}\sum_{t=[\tau_0 T]+1}^{T}\Big(y_{[\tau_0 T] }+\sum_{i=[\tau_0 T] +1}^{t-1}\varepsilon_i\Big)\varepsilon_t  \notag \\
&=&\frac{\beta_{1T}^{-[\tau_0 T]}}{\sqrt{T k_T} l(\eta_T)}\left(\Big(\beta_{1T}^{[\tau_0 T] }y_0+\sum_{j=1}^{[\tau_0 T] }\beta_{1T}^{[\tau_0 T]
-j}\varepsilon_j\Big)\sum_{t=[\tau_0 T] +1}^{T}\varepsilon_t+\sum_{t=[\tau_0 T] +1}^{T}\varepsilon_t\sum_{i=[\tau_0 T]+1}^{t-1}\varepsilon_i\right)  \notag \\
&:=&IV_1+IV_2,
\end{eqnarray}
where
\begin{eqnarray*}
IV_1=\frac{\beta_{1T}^{-[\tau_0 T]}}{\sqrt{T k_T} l(\eta_T)}\Big(\beta_{1T}^{[\tau_0 T] }y_0+\sum_{j=1}^{[\tau_0 T] }\beta_{1T}^{[\tau_0 T]-j}\varepsilon_j\Big)\sum_{t=[\tau_0 T] +1}^{T}\varepsilon_t
\end{eqnarray*}
and
\begin{eqnarray*}
IV_2=\frac{\beta_{1T}^{-[\tau_0 T]}}{\sqrt{T k_T} l(\eta_T)}\sum_{t=[\tau_0 T] +1}^{T}\varepsilon_t\sum_{i=[\tau_0 T] +1}^{t-1}\varepsilon_i.
\end{eqnarray*}

Applying part (b) of Lemma C.1 and assumption C3, it can be shown that
\begin{equation}  \label{D.1}
IV_{1}=\frac{1}{\sqrt{k_{T}l(\eta_{T})}}\sum_{j=1}^{[\tau_{0}T]}\beta_{1T}^{-j}\varepsilon_{j}\cdot \frac{1}{\sqrt{Tl(\eta_{T})}}\sum_{t=[\tau_{0}T]+1}^{T}\varepsilon_{t}+o_{p}(1)\Rightarrow Y(\overline{W}(1)-\overline{W}(\tau_{0})).
\end{equation}

In addition, it is not difficult to show that
\begin{equation}  \label{D.2}
IV_{2}=O_{p}\Big(\frac{\beta_{1T}^{-[\tau_{0}T]}}{\sqrt{Tk_{T}}l(\eta_{T})}\cdot Tl(\eta_{T})\Big)=o_{p}(1)
\end{equation}
by (C.1).

Combining (\ref{D.0}), (\ref{D.1}) and (\ref{D.2}) leads to part (a).

To prove part (b), we write
\begin{eqnarray}
&&\frac{\beta _{1T}^{-2[\tau _{0}T]}}{Tk_{T}l(\eta _{T})}\sum_{t=[\tau_{0}T]+1}^{T}y_{t-1}^{2}  \notag  \label{D.4} \\
&=&\frac{\beta _{1T}^{-2[\tau _{0}T]}}{Tk_{T}l(\eta _{T})}\sum_{t=[\tau_{0}T]+1}^{T}\Big(y_{[\tau _{0}T]}+\sum_{i=[\tau _{0}T]+1}^{t-1}\varepsilon
_{i}\Big)^{2}  \notag \\
&=&\frac{\beta _{1T}^{-2[\tau _{0}T]}}{Tk_{T}l(\eta _{T})}\sum_{t=[\tau_{0}T]+1}^{T}\Big(\sum_{j=1}^{[\tau _{0}T]}\beta _{1T}^{[\tau
_{0}T]-j}\varepsilon _{j}+\sum_{i=[\tau _{0}T]+1}^{t-1}\varepsilon _{i}\Big)^{2}\cdot (1+o_{p}(1))  \notag \\
&:=&(V_{1}+V_{2}+V_{3})\cdot (1+o_{p}(1)),
\end{eqnarray}
where
\begin{equation*}
V_{1}=\frac{1}{Tk_{T}l(\eta _{T})}\sum_{t=[\tau _{0}T]+1}^{T}\Big(\sum_{j=1}^{[\tau _{0}T]}\beta _{1T}^{-j}\varepsilon _{j}\Big)^{2},
\end{equation*}
\begin{equation*}
V_{2}=\frac{\beta _{1T}^{-2[\tau _{0}T]}}{Tk_{T}l(\eta _{T})}\sum_{t=[\tau_{0}T]+1}^{T}\Big(\sum_{i=[\tau _{0}T]+1}^{t-1}\varepsilon _{i}\Big)^{2}
\end{equation*}
and
\begin{equation*}
V_{3}=\frac{2\beta _{1T}^{-2[\tau _{0}T]}}{Tk_{T}l(\eta _{T})}\cdot\sum_{j=1}^{[\tau _{0}T]}\beta _{1T}^{[\tau _{0}T]-j}\varepsilon _{j}\cdot
\sum_{t=[\tau _{0}T]+1}^{T}\sum_{i=[\tau _{0}T]+1}^{t-1}\varepsilon_{i}
\end{equation*}

Obviously,
\begin{eqnarray}  \label{D.5}
V_1\Rightarrow (1-\tau_0)Y^2
\end{eqnarray}
by applying part (b) of Lemma C.1.

For the term $V_{2}$, we have
\begin{equation}  \label{D.6}
V_{2}=O_{p}\Big(\frac{\beta _{1T}^{-2[\tau _{0}T]}}{Tk_{T}l(\eta _{T})}\cdot T^{2}l(\eta _{T})\Big)=o_{p}(1)
\end{equation}
by making use of (C.1).

Finally, applying Cauchy-Schwarz inequality, it follows from (\ref{D.5}) and (\ref{D.6}) that
\begin{eqnarray}  \label{D.7}
V_3=o_p(1).
\end{eqnarray}

It is ready to see that part (b) is proved by combining (\ref{D.4})-(\ref{D.7}).

One can also verfiy that parts (a) and (b) hold jointly and $Y$ and $\overline{W}(1)-\overline{W}(\tau _{0})$ are mutually independent. $\hfill \Box $\newline



\noindent \textbf{Proof of Lemma D.3.} \ First, by applying Lemmas D.1 and D.2, it can be shown that
\begin{equation*}
A_{1}=o_{p}(1/k_{T})~~\mbox{and}~~A_{4}=o_{p}(1/k_{T}).
\end{equation*}
Secondly, we investigate the magnitude of the terms $A_{2}$ and $A_{5}$ in probability. For $A_{2}$, note that Cauchy-Schwarz inequality implies
\begin{equation*}
\sup_{m\in D_{1T}}\left\vert \frac{\sum_{t=m+1}^{[\tau_{0}T]}y_{t-1}\varepsilon _{t}}{\sum_{t=m+1}^{[\tau _{0}T]}y_{t-1}^{2}}\right\vert \leq \sup_{m\in D_{1T}}\sqrt{\frac{\sum_{t=m+1}^{[\tau_{0}T]}\varepsilon _{t}^{2}}{\sum_{t=m+1}^{[\tau _{0}T]}y_{t-1}^{2}}},
\end{equation*}
so it suffices to prove that the magnitude of the right hand side of the above inequality is $o_{p}(1/k_{T})$. Applying Lemma C.1, we immediately have
\begin{equation*}
\sup_{m\in D_{1T}}\sqrt{\frac{\sum_{t=m+1}^{[\tau _{0}T]}\varepsilon _{t}^{2}}{\sum_{t=1}^{[\tau _{0}T]}y_{t-1}^{2}}}\leq \sqrt{\frac{\sum_{t=1}^{[\tau_{0}T]}\varepsilon _{t}^{2}}{y_{[\tau _{0}T]-1}^{2}}}\leq O_{p}\left(\sqrt{\frac{Tl(\eta_{T})}{k_{T}\beta_{1T}^{2[\tau_{0}T]}l(\eta _{T})}}\right)
=o_{p}(1/k_{T})
\end{equation*}
by (C.1), which implies that
\begin{equation*}
A_{2}\leq \sup_{m\in D_{1T}}\sqrt{\frac{\sum_{t=m+1}^{[\tau_{0}T]}\varepsilon _{t}^{2}}{\sum_{t=1}^{[\tau _{0}T]}y_{t-1}^{2}}}=o_{p}(1/k_{T}).
\end{equation*}
Similarly, for the term $A_{5}$, we have
\begin{eqnarray*}
\sup_{m\in D_{2T}}\left\vert \frac{\sum_{t=[\tau_{0}T]+1}^{m}y_{t-1}\varepsilon _{t}}{\sum_{t=[\tau _{0}T]+1}^{m}y_{t-1}^{2}}\right\vert &\leq &\sup_{m\in D_{2T}}\sqrt{\frac{\sum_{t=[\tau_{0}T]+1}^{m}\varepsilon _{t}^{2}}{\sum_{t=[\tau _{0}T]+1}^{m}y_{t-1}^{2}}}\\
&\leq &\sqrt{\frac{\sum_{t=[\tau _{0}T]+1}^{T}\varepsilon _{t}^{2}}{y_{[\tau_{0}T]}^{2}}} \\
&=&O_{p}\left( \sqrt{\frac{Tl(\eta _{T})}{k_{T}\beta _{1T}^{2[\tau_{0}T]}l(\eta _{T})}}\right) \\
&=&o_{p}(1/k_{T}),
\end{eqnarray*}
which implies that
\begin{equation*}
A_{5}=o_{p}(1/k_{T}).
\end{equation*}
Thirdly, note also that
\begin{eqnarray*}
A_{3} &=&\sup_{m\in D_{1T}}\Big|\frac{\sum_{t=m+1}^{T}y_{t-1}^{2}}{\sum_{t=[\tau _{0}T]+1}^{T}y_{t-1}^{2}\sum_{t=m+1}^{[\tau _{0}T]}y_{t-1}^{2}}\Lambda _{T}(\frac{m}{T})\Big| \\
&\leq &\bigg(\frac{1}{\sum_{t=[\tau _{0}T]+1}^{T}y_{t-1}^{2}}+\frac{1}{\sum_{t=[\tau _{0}T]-M_{T}}^{[\tau _{0}T]}y_{t-1}^{2}}\bigg)\bigg(\sup_{m\in
D_{1T}}\Big|\frac{(\sum_{t=1}^{[\tau _{0}T]}y_{t-1}\varepsilon _{t})^{2}}{\sum_{t=1}^{[\tau _{0}T]}y_{t-1}^{2}}-\frac{(\sum_{t=1}^{m}y_{t-1}\varepsilon _{t})^{2}}{\sum_{t=1}^{m}y_{t-1}^{2}}\Big| \\
&&~~~~+\sup_{m\in D_{1T}}\Big|\frac{(\sum_{t=[\tau_{0}T]+1}^{T}y_{t-1}\varepsilon _{t})^{2}}{\sum_{t=[\tau_{0}T]+1}^{T}y_{t-1}^{2}}-\frac{(\sum_{t=m+1}^{T}y_{t-1}\varepsilon_{t})^{2}}{\sum_{t=m+1}^{T}y_{t-1}^{2}}\Big|\bigg) \\
&\leq &\Big(O_{p}\big(\frac{1}{Tk_{T}\beta _{1T}^{2[\tau _{0}T]}l(\eta_{T})}\big)+O_{p}\big(\frac{1}{k_{T}\beta _{1T}^{2[\tau _{0}T]}l(\eta _{T})}\big)\Big)\cdot O_{p}(l(\eta_{T})) \\
&=&o_{p}(1/k_{T}^{2}),
\end{eqnarray*}
and
\begin{eqnarray*}
A_{6} &=&\sup_{m\in D_{2T}}\Big|\frac{\sum_{t=1}^{m}y_{t-1}^{2}}{\sum_{t=[\tau _{0}T]+1}^{m}y_{t-1}^{2}\sum_{t=1}^{[\tau _{0}T]}y_{t-1}^{2}}\Lambda _{T}(\frac{m}{T})\Big| \\
&\leq &\bigg(\frac{1}{\sum_{t=1}^{[\tau _{0}T]}y_{t-1}^{2}}+\frac{1}{\sum_{t=[\tau _{0}T]+1}^{[\tau _{0}T]+M_{T}}y_{t-1}^{2}}\bigg)\bigg(\sup_{m\in D_{2T}}\Big|\frac{(\sum_{t=1}^{[\tau _{0}T]}y_{t-1}\varepsilon_{t})^{2}}{\sum_{t=1}^{[\tau _{0}T]}y_{t-1}^{2}}-\frac{(\sum_{t=1}^{m}y_{t-1}\varepsilon _{t})^{2}}{\sum_{t=1}^{m}y_{t-1}^{2}}\Big|\\
&&~~~~+\sup_{m\in D_{2T}}\Big|\frac{(\sum_{t=[\tau_{0}T]+1}^{T}y_{t-1}\varepsilon _{t})^{2}}{\sum_{t=[\tau_{0}T]+1}^{T}y_{t-1}^{2}}-\frac{(\sum_{t=m+1}^{T}y_{t-1}\varepsilon _{t})^{2}}{\sum_{t=m+1}^{T}y_{t-1}^{2}}\Big|\bigg) \\
&\leq &\Big(O_{p}(\frac{1}{k_{T}^{2}\beta _{1T}^{2[\tau _{0}T]}l(\eta _{T})})+O_{p}(\frac{1}{k_{T}\beta _{1T}^{2[\tau _{0}T]}l(\eta _{T})})\Big)\cdot
O_{p}(l(\eta _{T})) \\
&=&o_{p}(1/k_{T}^{2}).
\end{eqnarray*}
These complete the proofs. $\hfill \Box $\newline


\noindent \textbf{Proof of Lemma D.4.} \ To prove part (a), applying equation (B.2) in Chong (2001), we have
\begin{eqnarray*}
&&RSS_{T}(\tau _{0}-\frac{m}{T})-RSS_{T}(\tau _{0}) \\
&=&2(\beta _{2}-\beta _{1T})\bigg(\frac{\sum_{t=[\tau _{0}T]-m+1}^{[\tau_{0}T]}y_{t-1}^{2}\sum_{t=[\tau _{0}T]+1}^{T}y_{t-1}\varepsilon_{t}}{\sum_{t=[\tau _{0}T]-m+1}^{T}y_{t-1}^{2}}-\frac{\sum_{t=[\tau_{0}T]+1}^{T}y_{t-1}^{2}\sum_{t=[\tau _{0}T]-m+1}^{[\tau_{0}T]}y_{t-1}\varepsilon _{t}}{\sum_{t=[\tau _{0}T]-m+1}^{T}y_{t-1}^{2}}\bigg) \\
&&~+(\beta _{2}-\beta _{1T})^{2}\frac{\sum_{t=[\tau_{0}T]+1}^{T}y_{t-1}^{2}\sum_{t=[\tau _{0}T]-m+1}^{[\tau _{0}T]}y_{t-1}^{2}}{\sum_{t=[\tau_{0}T]-m+1}^{T}y_{t-1}^{2}}+\Lambda _{T}(\tau _{0}-\frac{m}{T}),
\end{eqnarray*}
where
\begin{eqnarray*}
&&\Lambda _{T}(\tau _{0}-\frac{m}{T}) \\
&=&\frac{\Big(\sum_{t=1}^{[\tau _{0}T]}y_{t-1}\varepsilon _{t}\Big)^{2}}{\sum_{t=1}^{[\tau _{0}T]}y_{t-1}^{2}}-\frac{\Big(\sum_{t=1}^{[\tau_{0}T]-m}y_{t-1}\varepsilon _{t}\Big)^{2}}{\sum_{t=1}^{[\tau_{0}T]-m}y_{t-1}^{2}}+\frac{\Big(\sum_{t=[\tau_{0}T]+1}^{T}y_{t-1}\varepsilon _{t}\Big)^{2}}{\sum_{t=[\tau_{0}T]+1}^{T}y_{t-1}^{2}}-\frac{\Big(\sum_{t=[\tau
_{0}T]-m+1}^{T}y_{t-1}\varepsilon _{t}\Big)^{2}}{\sum_{t=[\tau_{0}T]-m+1}^{T}y_{t-1}^{2}}.
\end{eqnarray*}
Obviously, $\Lambda_{T}(\tau _{0}-\frac{m}{T})=O_{p}(l(\eta_{T}))$. It follows that
\begin{eqnarray*}
&&\frac{k_{T}}{\beta_{1T}^{2[\tau _{0}T]}l(\eta_{T})}\Big(RSS_{T}(\tau_{0}-\frac{m}{T})-RSS_{T}(\tau _{0})\Big) \\
&=&\frac{k_{T}}{\beta_{1T}^{2[\tau _{0}T]}l(\eta_{T})}\cdot \Big(-\frac{2c}{k_{T}}\Big)\bigg(\frac{O_{p}(mk_{T}\beta _{1T}^{2[\tau _{0}T]}l(\eta
_{T}))O_{p}(\sqrt{Tk_{T}}\beta_{1T}^{[\tau_{0}T]}l(\eta _{T}))}{O_{p}(mk_{T}\beta _{1T}^{2[\tau_{0}T]}l(\eta _{T}))+O_{p}(Tk_{T}\beta_{1T}^{2[\tau _{0}T]}l(\eta _{T}))}+\frac{O_{p}(Tk_{T}\beta_{1T}^{2[\tau_{0}T]}l(\eta _{T}))}{O_{p}(\sqrt{k_{T}}\beta_{1T}^{[\tau_{0}T]})}\bigg) \\
&&+\frac{c^{2}}{k_{T}\beta_{1T}^{2[\tau _{0}T]}l(\eta _{T})}\cdot \frac{O_{p}(Tk_{T}\beta _{1T}^{2[\tau _{0}T]}l(\eta _{T}))}{O_{p}(mk_{T}\beta_{1T}^{2[\tau _{0}T]}l(\eta _{T}))+O_{p}(Tk_{T}\beta_{1T}^{2[\tau_{0}T]}l(\eta _{T}))}\sum_{[\tau_{0}T]-m+1}^{[\tau_{0}T]}y_{t-1}^{2}+o_{p}(1) \\
&=&\frac{c^{2}(1+o_{p}(1))}{k_{T}\beta _{1T}^{2[\tau _{0}T]}l(\eta _{T})}\sum_{[\tau _{0}T]-m+1}^{[\tau _{0}T]}y_{t-1}^{2}+o_{p}(1) \\
&\Rightarrow &c^{2}mY^{2}
\end{eqnarray*}%
by Lemmas C.1 and D.2 and the observations
\begin{equation*}
\bigg|\frac{\sum_{t=[\tau _{0}T]-m+1}^{[\tau _{0}T]}y_{t-1}\varepsilon_{t}}{\sum_{t=[\tau _{0}T]-m+1}^{T}y_{t-1}^{2}}\bigg|\leq \frac{\sqrt{\sum_{t=[\tau _{0}T]-m+1}^{[\tau _{0}T]}\varepsilon _{t}^{2}}}{\sqrt{\sum_{t=[\tau _{0}T]-m+1}^{T}y_{t-1}^{2}}}=O_{p}(\frac{1}{\sqrt{k_{T}}\beta_{1T}^{[\tau_{0}T]}})
\end{equation*}
and
\begin{equation*}
\frac{\sum_{t=[\tau _{0}T]-m+1}^{[\tau _{0}T]}y_{t-1}^{2}}{my_{[\tau_{0}T]}^{2}}\overset{p}{\rightarrow } 1.
\end{equation*}

To prove part (b), applying equation (B.4) in Chong (2001), we have
\begin{eqnarray*}
&&RSS_{T}(\tau _{0}+\frac{m}{T})-RSS_{T}(\tau_{0}) \\
&=&2(\beta _{2}-\beta _{1T})\bigg(\frac{\sum_{t=1}^{[\tau_{0}T]}y_{t-1}^{2}\sum_{t=[\tau _{0}T]+1}^{[\tau _{0}T]+m}y_{t-1}\varepsilon
_{t}}{\sum_{t=1}^{[\tau _{0}T]+m}y_{t-1}^{2}}-\frac{\sum_{t=[\tau_{0}T]+1}^{[\tau _{0}T]+m}y_{t-1}^{2}\sum_{t=1}^{[\tau_{0}T]}y_{t-1}\varepsilon _{t}}{\sum_{t=1}^{[\tau _{0}T]+m}y_{t-1}^{2}}\bigg)\\
&&~+(\beta_{2}-\beta_{1T})^{2}\frac{\sum_{t=[\tau_{0}T]+1}^{[\tau_{0}T]+m}y_{t-1}^{2}\sum_{t=1}^{[\tau_{0}T]}y_{t-1}^{2}}{\sum_{t=1}^{[\tau_{0}T]+m}y_{t-1}^{2}}+\Lambda _{T}(\tau _{0}+\frac{m}{T})
\end{eqnarray*}
where
\begin{eqnarray*}
&&\Lambda _{T}(\tau _{0}+\frac{m}{T}) \\
&=&\frac{\Big(\sum_{t=1}^{[\tau _{0}T]}y_{t-1}\varepsilon _{t}\Big)^{2}}{\sum_{t=1}^{[\tau _{0}T]}y_{t-1}^{2}}-\frac{\Big(\sum_{t=1}^{[\tau
_{0}T]+m}y_{t-1}\varepsilon _{t}\Big)^{2}}{\sum_{t=1}^{[\tau_{0}T]+m}y_{t-1}^{2}}+\frac{\Big(\sum_{t=[\tau
_{0}T]+1}^{T}y_{t-1}\varepsilon _{t}\Big)^{2}}{\sum_{t=[\tau_{0}T]+1}^{T}y_{t-1}^{2}}-\frac{\Big(\sum_{t=[\tau
_{0}T]+m+1}^{T}y_{t-1}\varepsilon _{t}\Big)^{2}}{\sum_{t=[\tau_{0}T]+m+1}^{T}y_{t-1}^{2}}.
\end{eqnarray*}
Then, using the facts $\Lambda _{T}(\tau _{0}+\frac{m}{T})=O_{p}(l(\eta_{T}))$,
\begin{equation*}
\bigg|\frac{\sum_{t=[\tau_{0}T]+1}^{[\tau_{0}T]+m}y_{t-1}\varepsilon_{t}}{\sum_{t=[\tau _{0}T]+1}^{[\tau _{0}T]+m}y_{t-1}^{2}}\bigg|\leq \frac{\sqrt{\sum_{t=[\tau_{0}T]+1}^{[\tau_{0}T]+m}\varepsilon_{t}^{2}}}{\sqrt{\sum_{t=[\tau_{0}T]+1}^{[\tau_{0}T]+m}y_{t-1}^{2}}}=O_{p}(\frac{1}{\sqrt{k_{T}}\beta_{1T}^{[\tau_{0}T]}}),
\end{equation*}
and
\begin{equation*}
\frac{\sum_{t=[\tau _{0}T]+1}^{[\tau _{0}T]+m}y_{t-1}^{2}}{my_{[\tau_{0}T]}^{2}}\overset{p}{\rightarrow } 1,
\end{equation*}
we have
\begin{eqnarray*}
&&\frac{k_{T}}{\beta_{1T}^{2[\tau _{0}T]}l(\eta_{T})}\Big(RSS_{T}(\tau_{0}+\frac{m}{T})-RSS_{T}(\tau _{0})\Big) \\
&=&\frac{k_{T}}{\beta_{1T}^{2[\tau _{0}T]}l(\eta_{T})}\cdot \Big(-\frac{2c}{k_{T}}\Big)\bigg(\frac{O_{p}(k_{T}^{2}\beta_{1T}^{2[\tau_{0}T]}l(\eta
_{T}))}{\sqrt{k_{T}}\beta_{1T}^{[\tau _{0}T]}}+\frac{O_{p}(mk_{T}\beta_{1T}^{2[\tau _{0}T]}l(\eta _{T}))O_{p}(k_{T}\beta_{1T}^{[\tau
_{0}T]}l(\eta _{T}))}{O_{p}(k_{T}^{2}\beta_{1T}^{2[\tau_{0}T]}l(\eta_{T}))+O_{p}(mk_{T}\beta _{1T}^{2[\tau _{0}T]}l(\eta_{T}))}\bigg) \\
&&+\frac{c^{2}}{k_{T}\beta_{1T}^{2[\tau_{0}T]}l(\eta_{T})}\cdot \frac{O_{p}(k_{T}^{2}\beta _{1T}^{2[\tau _{0}T]}l(\eta_{T}))}{O_{p}(k_{T}^{2}\beta _{1T}^{2[\tau _{0}T]}l(\eta_{T}))+O_{p}(mk_{T}\beta_{1T}^{2[\tau_{0}T]}l(\eta _{T}))}\sum_{t=[\tau_{0}T]+1}^{[\tau_{0}T]+m}y_{t-1}^{2}+o_{p}(1) \\
&=&\frac{c^{2}(1+o_{p}(1))}{k_{T}\beta_{1T}^{2[\tau_{0}T]}l(\eta_{T})}\sum_{t=[\tau _{0}T]+1}^{[\tau_{0}T]+m}y_{t-1}^{2}+o_{p}(1) \\
&\Rightarrow &c^{2}mY^{2}
\end{eqnarray*}
by Lemmas C.1 and D.1. $\hfill \Box $\newline

\bigskip

\begin{thebibliography}{99}

\bibitem{} {\footnotesize \textsc{Cs\"{o}rg\H{o}, M., Szyszkowicz, B. and Wang, Q. Y.} (2003). Donsker's theorem for self-normalized partial sums
processes. \emph{Annals of Probability} \textbf{31} (3): 1228-1240. }


\bibitem{} {\footnotesize \textsc{Gin\'{e}, E., G\"{o}tze, F. and Mason, D. M.} (1997). When is the Student $t$-statistic asymptotically standard
normal? \emph{Annals of Probability} \textbf{25} (3): 1514--1531. }


\bibitem{} {\footnotesize \textsc{Huang S. H., Pang, T. X. and Weng, C. G.} (2014). Limit theory for moderate deviations from a unit root under
innovations with a possibly infinite variance. \emph{Methodology and Computing in Applied Probability } \textbf{16} (1): 187-206. }

\bibitem{} {\footnotesize \textsc{Phillips, P. C. B. and Magdalinos, T.} (2007a). Limit theory for moderate deviations from a unit root. \emph{Journal of Econometrics} \textbf{136}: 115--130. }

\end{thebibliography}

\end{document}