\documentclass[a4paper,11pt]{article}
\usepackage{epsfig,amsmath,amssymb,amsthm}

\usepackage{epsfig,amsmath,amssymb,amsthm}

\usepackage{openbib}
\usepackage[round,longnamesfirst]{natbib}
\usepackage[ansinew]{inputenc}
\usepackage{graphicx}
%\usepackage{multirow}
%\usepackage{diagbox}
%\usepackage{harvard}

\usepackage{enumerate}
\usepackage{fullpage}
\usepackage{setspace}
\usepackage{rotating}
\usepackage{dsfont}
\usepackage{mathrsfs}
\usepackage[ansinew]{inputenc}
%\usepackage{KendallsTau-abbreviations}
\usepackage{verbatim,color}
%--------------------------------------------------
% Verweise auf Lemata im main text
\usepackage{xr}
\externaldocument{DVWW-main}
%\externaldocument{DVWW-appendix}

% ========================================================================
% definitions & abbreviations
% ========================================================================


\setcounter{secnumdepth}{3} \setcounter{tocdepth}{3}
\renewcommand{\baselinestretch}{1}
\relax

\DeclareMathOperator{\cov}{Cov}

\newtheorem{lemma}{Lemma}[section]
\newtheorem{theorem}[lemma]{Theorem}
\newtheorem{proposition}[lemma]{Proposition}
\newtheorem{definition}[lemma]{Definition}
\newtheorem{corollary}[lemma]{Corollary}
\newtheorem{example}[lemma]{Example}
\newtheorem{remark}[lemma]{Remark}
\newtheorem{model}[lemma]{Model}
\newtheorem{assumption}[lemma]{Assumption}

% calligraphic and double stroke  -----------------------------------------

\newcommand{\calA}{\mathcal{A}}
\newcommand{\calB}{\mathcal{B}}
\newcommand{\Ee}{\mathscr{E}} 
\newcommand{\F}{\mathscr{F}}
\newcommand{\dsF}{\mathds{F}} 
\newcommand{\G}{\mathscr{G}}
\newcommand{\N}{\mathds{N}}  
\newcommand{\Pb}{\mathds{P}}   
\newcommand{\R}{\mathds{R}}   
\newcommand{\Z}{{\mathds{Z}}} 

% bold mathematical symbols ------------------------------------------------

\newcommand{\bx}{\boldsymbol{x}}
\newcommand{\by}{\boldsymbol{y}}
\newcommand{\bz}{\boldsymbol{z}}
\newcommand{\bK}{\boldsymbol{K}}
\newcommand{\bU}{\boldsymbol{U}}
\newcommand{\bX}{\boldsymbol{X}}
\newcommand{\bY}{\boldsymbol{Y}}
\newcommand{\bZ}{\boldsymbol{Z}}

\newcommand{\bNull}{\mbox{\boldmath{$0$}}}
\newcommand{\bEins}{\mbox{\boldmath{$1$}}}

% variance and covariance --------------------------------------------------

\newcommand{\Var}{\mathrm{Var}}
\newcommand{\Cov}{\mathrm{Cov}}
\newcommand{\Cor}{\mathrm{Cor}}

% convergence arrows, relations  --------------------------------------------

%\newcommand{\cid}{\stackrel{\mbox{\tiny $\mathscr{L}$}}{\longrightarrow}} %Convergence In Distribution
\newcommand{\cid}{\stackrel{\mbox{\tiny d}}{\longrightarrow}} %Convergence In Distribution
\newcommand{\cip}{\stackrel{\mbox{\tiny $p$}}{\longrightarrow}} %Convergence In Probability
\newcommand{\asc}{\stackrel{\mbox{\tiny $a.s.$}}{\longrightarrow}} %Almost Sure Convergence
\newcommand{\claw}{{\stackrel{\mbox{\tiny d}}{\longrightarrow}}}

% environments -------------------------------------------------------------

\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}

\newcommand{\ben}{\begin{enumerate}[a)]} % package 'enumerate' 
\newcommand{\bEN}{\begin{enumerate}[1.]}
\newcommand{\een}{\end{enumerate}}
\newcommand{\eEN}{\end{enumerate}}
\newcommand{\bit}{\begin{itemize}}
\newcommand{\eit}{\end{itemize}}

% mathematical operators ---------------------------------------------------

\DeclareMathOperator*{\argmin}{arg\,min}
\DeclareMathOperator*{\argmax}{arg\,max}
\DeclareMathOperator*{\esssup}{ess\,sup}


% sections and subsections style --------------------------------------------

\setlength{\baselineskip}{0.70cm}

\parindent 0cm

\makeatletter
\def\@seccntformat#1{\csname the#1\endcsname. }
\def\section{\@startsection {section}{1}{\z@}{-3.5ex plus -1ex minus
    -.2ex}{1.3ex plus .2ex}{\center\large\sc}}
\def\subsection{\@startsection{subsection}{2}{\z@}{3.25ex plus 1ex minus .2ex}{-1em}{\normalsize\bf}}

\def\subsubsection{\@startsection{subsubsection}{3}{\z@}{3.25ex plus 1ex minus .2ex}{-1em}{\normalsize\it}}




% ========================================================================
% title information
% ========================================================================

%\setcounter{secnumdepth}{3} \setcounter{tocdepth}{3}
%\renewcommand{\baselinestretch}{1}
%\relax
%\newtheorem{lemma}{Lemma}[section]
%\DeclareMathOperator{\cov}{Cov}
%

\markright{Online supplement}


\title{\textsc{Supplementary Material on \\ ``Testing for changes in Kendall's tau''}}

\author{
	\textsc{Herold Dehling} \\
	\textit{\small University of Bochum} \\[3.0ex]
	\textsc{Daniel Vogel}\footnote{Corresponding author: Daniel Vogel, E-Mail: daniel.vogel@abdn.ac.uk.} \\
	 \textit{\small University of Aberdeen} \\[3.0ex]
	\textsc{Martin Wendler} \\ 
	 \textit{\small University of Greifswald}\\[3.0ex]
		\textsc{Dominik Wied} \\
 \textit{\small  University of Cologne}% \\[3.0ex]
}
\date{}

%---------------------------------------------------------------------------------------------------------------------------------------------------------------
\begin{document}

%\setlength{\baselineskip}{0.70cm}
%
%\parindent 0cm
%
%\makeatletter
%\def\@seccntformat#1{\csname the#1\endcsname. }
%\def\section{\@startsection {section}{1}{\z@}{-3.5ex plus -1ex minus
    %-.2ex}{1.3ex plus .2ex}{\center\large\sc}}
%\def\subsection{\@startsection{subsection}{2}{\z@}{3.25ex plus 1ex minus .2ex}{-1em}{\normalsize\bf}}
%
%\def\subsubsection{\@startsection{subsubsection}{3}{\z@}{3.25ex plus 1ex minus .2ex}{-1em}{\normalsize\it}}
%


\maketitle
%\newpage
%
%{\bf Running Head:} Online supplement
%%\end{center}
%%\begin{center}
%
%\bigskip
%{\bf Proofs should be sent to:} Daniel Vogel, daniel.vogel@abdn.ac.uk

%\end{center}

\bigskip
\begin{center}
{\bf Abstract}
\end{center}

This document contains proofs and further technical results for the article ``Testing for changes in Kendall's tau''.
It consists of one section, labeled Appendix D. The results are labeled D.1, D.2, $\ldots$. Results from the main document are referred to as in the main document. The labels contain references to the respective section, e.g., Corollary 3.1 can be found in Section 3, Lemma A.2 in Appendix A of the main document.




%\newpage
\appendix
\setcounter{section}{3}
\section{Proofs of lemmas}


%%%%--------------------------------------------------------------------------------------------------------------------------------------------------
%%%===================================================================================================================================================
% Lemma A1

%%%%--------------------------------------------------------------------------------------------------------------------------------------------------
%%%===================================================================================================================================================

We first prove Lemmas \ref{lem:l_0} and \ref{lem:neu1} from Appendix \ref{sec:app:pned}.


\begin{proof}[Proof of Lemma \ref{lem:l_0}]
Part (\ref{lem:l_0(i)}) is straightforward. \par
\emph{Part (\ref{lem:l_0(ii)}):} There are positive constants $C_1, C_2$ such that
\begin{eqnarray}
%\nonumber
a^p_{p,k} 			\ = \ 			E\left| \bX_0 - E\left( \bX_0 \middle| \F_{-k}^k \right) \right|_p^p 
\nonumber		& \le & C_1 E\left| \bX_0 - E\left( \bX_0 \middle| \F_{-k}^k \right) \right|_1 \\
\label{eq:2}		& \le & C_2 E\left| \bX_0 - f_{k}\left(\bZ_{-k},\ldots,\bZ_k \right) \right|_1.
\end{eqnarray}
The first inequality is due to the boundedness of $\bX_0$. The second inequality can be shown by applying Jensen's inequality for the conditional expectation $E(\cdot | \F_{-k}^k  )$ to the convex function $|\cdot|_1$. Further, for any $\varepsilon > 0$ we have
\[
	E\left| \bX_0 - f_{k}\left(\bZ_{-k},\ldots,\bZ_k \right) \right|_1
	\ \le \ 
	C_3 P\left( \left| \bX_0 - f_{k}\left(\bZ_{-k},\ldots,\bZ_k \right) \right|_1 > \varepsilon \right)
 +   \varepsilon
	\ \le \  
	C_3 \Phi(\varepsilon) a_k
	 +   \varepsilon
\]
Combining this with (\ref{eq:2}) we arrive at $a^p_{p,k}  \le  C_2 C_3 \Phi(\varepsilon) a_k  +   C_2 \varepsilon$.
By first choosing $\varepsilon$ sufficiently small and then $k$ sufficiently large we can make the left-hand side arbitrarily small, and hence $(\bX_n)_{n \in \Z}$ is $L_p$-NED, $p \ge 1$, on $(\bZ_n)_{n \in \Z}$.
In particular, if condition (\ref{eq:ned.con}) holds, we get 
$a^p_{p,k}  \le  C_2 C_3 \Phi(s_k) a_k  +   C_2 s_k  =  O(s_k)$.
%
%by taking $\varepsilon = s_k$:
%
%
\emph{Part (\ref{lem:l_0(iii)}):}
Let $f_k(\bZ_{-k},\ldots,\bZ_k) = E(\bX_0|\F_{-k}^k)$. By means of the H\"older and the Markov inequalities we have for every $\varepsilon > 0$:
\[
	P\left( \left| \bX_0 - f_{k}\left(\bZ_{-k},\ldots,\bZ_k \right) \right|_1 > \varepsilon \right)
	\, \le \,
  	q^{p-1} \varepsilon^{-p }\, E\left| \bX_0 - E(\bX_0|\F_{-k}^k) \right|_p^p 
\]
Choosing $\Phi(\varepsilon) = q^{p-1} \varepsilon^{-p}$ and $a_k = a_{p,k}^p$ we have $a_k \to 0$ as $k \to \infty$, and $(\bX_n)_{n \in \Z}$ is hence $P$-NED on $(\bZ_n)_{n \in \Z}$.
\end{proof}


%%%%--------------------------------------------------------------------------------------------------------------------------------------------------
%%%===================================================================================================================================================
% Lemma A2

%%%%--------------------------------------------------------------------------------------------------------------------------------------------------
%%%===================================================================================================================================================

\begin{proof}[Proof of Lemma \ref{lem:neu1}] By the definition of $g_1$, we have that, for any $\bx,\bx'\in\R^d$,
\begin{equation*}
\left|g_1(\bx)-g_1(\bx')\right|=\left|Eg(\bx,\bX_0)-Eg(\bx',\bX_0)\right|\leq \int\left|g(\bx,\by)-g(\bx',\by)\right|dF(\by)
\end{equation*}
and consequently, for independent copies $\bX,\bY$ of $\bX_0$,
\begin{multline*}
E\left(\sup_{|\bx-\bX|\leq\epsilon}\left|g_1(\bx)-g_1(\bX)\right|\right)^2 \\
\leq E\left(\sup_{|\bx-\bX|\leq\epsilon}\int\left|g(\bx,\by)-g(\bX,\by)\right|dF(\by)\right)^2
\leq E\left(\int\sup_{|\bx-\bX|\leq\epsilon}\left|g(\bx,\by)-g(\bX,\by)\right|dF(\by)\right)^2 \\
\leq E\left(\sup_{|\bx-\bX|\leq\epsilon}\left|g(\bx,\bY)-g(\bX,\bY)\right|\right)^2
\leq E\left(\sup_{|\bx-\bX|\leq\epsilon,|\by-\bY|\leq\epsilon}\left|g(\bx,\by)-g(\bX,\bY)\right|\right)^2\leq L\epsilon.
\end{multline*}
Recall that the conditional expectation minimizes the $L_2$-distance, so
\begin{equation*}
	E \left\{ g_1(\bX_0) - E (g_1(\bX_0) | \F^k_{-k}) \right\}^2 
	\leq 
	E \left\{ g_1(\bX_0) - g_1(\bX_{0,k}) \right\}^2,
\end{equation*}
where $\bX_{0,k} = f_k(\bZ_{-k},\ldots, \bZ_k)$. 
Now we will make use of the $P$-near epoch dependence and the H\"older inequality and obtain
\begin{multline*}
E \left( g_1(\bX_0) - g_1(\bX_{0,k}) \right)^2\\
=  E \left( g_1(\bX_0) - g_1(\bX_{0,k}) \right)^2 \bEins_{\{|\bX_0-\bX_{0,k}|>s_l\}}
	+ E\left( g_1(\bX_0) - g_1(\bX_{0,k}) \right)^2 \bEins_{\{|\bX_0-\bX_{0,k}|\leq s_l\} }\\
\leq 
\left(2\left\|g_1^2(\bX_0)\right\|_{\frac{2+\delta}{2}}+2\left\|g_1^2(\bX_{0,k})\right\|_{\frac{2+\delta}{2}}\right)
		\left(P(|\bX_0-\bX_{0,k}|>s_l)\right)^{\frac{\delta}{2+\delta}} \\
+ E\left( g_1(\bX_0) - g_1(\bX_{0,k}) \right)^2 \bEins_{\{|\bX_0-\bX_{0,k}|\leq s_l\}}
\leq C(s_l^{\frac{2+\delta}{\delta}})^{\frac{\delta}{2+\delta}}+ L s_l
\leq Cs_l
\end{multline*}
and finally
\begin{equation*}
\left\|g_1(\bX_0) - E (g_1(\bX_0) | \F^k_{- k})\right\|_2\leq  \left(E \left( g_1(\bX_0) - g_1(\bX_{0,k}) \right)^2\right)^{1/2}\leq Cs_l^{1/2},
\end{equation*}
which completes the proof.
\end{proof}
%




%%%%--------------------------------------------------------------------------------------------------------------------------------------------------
%%%===================================================================================================================================================
% das war mal Appendix B

%%%%--------------------------------------------------------------------------------------------------------------------------------------------------
%%%===================================================================================================================================================

The following lemmas \ref{lem2} to \ref{lem6} are required for the proof of Theorem \ref{theo1} (Invariance principle for the sequential $U$-process).




\begin{lemma}\label{lem2} Let $(\bZ_n)_{n\in\Z}$ be a stationary and absolutely regular stochastic process with mixing coefficients $(\beta_k)_{k\in\N}$. Then there exist processes $(\bZ'_n)_{n\in\Z}$ and $(\bZ''_n)_{n\in\Z}$, independent of each other and both with the same distribution as $(\bZ_n)_{n\in\Z}$ such that
$P\{(\bZ'_n)_{n\geq k}=(\bZ_n)_{n\geq k}\}=1-\beta_k$ and $P\{(\bZ''_n)_{n\leq 0}=(\bZ_n)_{n\leq 0}\}=1-\beta_k$.
%\begin{enumerate}
%\item $P\left((\bZ'_n)_{n\geq k}=(\bZ_n)_{n\geq k}\right)=1-\beta_k$.
%\item $P\left((\bZ''_n)_{n\leq 0}=(\bZ_n)_{n\leq 0}\right)=1-\beta_k$.
%\end{enumerate}
\end{lemma}
%
This can be proved in a similar way to Lemma 2.5 of \citet{borovkova:burton:dehling:2001}.


%
%
%
%
%
%
%
\begin{lemma}\label{lem3} Let $(\bX_n)_{n\in\Z}$ be $P$-NED of an absolutely regular sequence $(\bZ_n)_{n\in\Z}$ and $g$ be a kernel, such that Assumptions \ref{ass1} and \ref{ass2} hold. Then there is a constant $C>0$, such that for $i,k,l\in\N$, $\epsilon>0$:
\begin{equation*}
	\left\|g_2(\bX_i,\bX_{i+k+2l})-g_2(\bX_{i,l},\bX_{i+k+2l,l})\right\|_2
	\leq 
	C (\sqrt{\epsilon}+a_l^{\frac{\delta}{2+\delta}}\Phi^{\frac{\delta}{2+\delta}}(\epsilon)+\beta^{\frac{\delta}{2+\delta}}_k), 
\end{equation*}
where $\bX_{i,l}=f_l(\bZ_{i-l},\ldots,\bZ_{i+l})$.
\end{lemma}
%
%
\begin{proof}[Proof of Lemma \ref{lem3}] First note that we can rewrite any $\bX_i$ such that
\begin{equation*}
	\bX_i	=	f_\infty\left((\bZ_{i+n})_{n\in\Z}\right).
\end{equation*}
By Lemma \ref{lem2} there exist independent copies $(\bZ'_n)_{n\in\Z}$ and $(\bZ''_n)_{n\in\Z}$ of $(\bZ_n)_{n\in\Z}$ satisfying
$P\{(\bZ'_n)_{n\geq i+l+k}=(Z_n)_{n\geq i+l+k}\}=1-\beta_k$ and $P\{(\bZ''_n)_{n\leq i+l}=(Z_n)_{n\leq i+l}\}=1-\beta_k$.
We define $\bX'_i=f_{\infty}((\bZ'_{n+i})_{n\in\Z})$ and $\bX''_i=f_{\infty}((\bZ''_{n+i})_{n\in\Z})$.
%
%
Analogously, $\bX'_{i,l}=f_{l}(\bZ'_{i-l},\ldots,\bZ'_{i+l})$ and $\bX''_{i,l}=f_{l}(\bZ'_{i-l},\ldots,\bZ'_{i+l})$.  Now we can conclude that
\begin{multline*}
	\left\|g_2(\bX_i,\bX_{i+k+2l})-g_2(\bX_{i,l},\bX_{i+k+2l,l})\right\|_2 
	\leq \left\|g_2(\bX_i,\bX_{i+k+2l})-g_2(\bX''_{i},\bX'_{i+k+2l})\right\|_2 \\
	+	\left\|g_2(\bX''_i,\bX'_{i+k+2l})-g_2(\bX''_{i,l},\bX'_{i+k+2l,l})\right\|_2 
	+ \left\|g_2(\bX''_{i,l},\bX'_{i+k+2l,l})-g_2(\bX_{i,l},\bX_{i+k+2l,l})\right\|_2 
\end{multline*}
We will treat the three summands on the right-hand side separately. For the first summand, we use the H\"older inequality to obtain
\begin{multline*}
	\left\|g_2(\bX_i,\bX_{i+k+2l})-g_2(\bX''_{i},\bX'_{i+k+2l})\right\|_2\\
	= \big\| \{g_2(\bX_i,\bX_{i+k+2l})-g_2(\bX''_{i},\bX'_{i+k+2l})\}
	\bEins_{\left\{(\bZ'_n)_{n\geq i+l+k}\neq(\bZ_n)_{n\geq i+l+k}\text{ or } (\bZ''_n)_{n\leq i+l}\neq(\bZ_n)_{n\leq i+l}\right\}} \big\|_2 \\
	\leq 	
	\left\|(g_2(\bX_i,\bX_{i+k+2l})-g_2(\bX''_{i},\bX'_{i+k+2l})\right\|_{\frac{2+\delta}{2}}\\ 
\times \left(P\{(\bZ'_n)_{n\geq i+l+k}\neq(\bZ_n)_{n\geq i+l+k}\text{ or } (\bZ''_n)_{n\leq i+l}\neq(\bZ_n)_{n\leq i+l}\}\right)^{\frac{\delta}{2+\delta}}
	\leq 2M^{2/(2+\delta)}2\beta^{\delta/(2+\delta)}_k
\end{multline*}
with Assumption \ref{ass2} and Lemma \ref{lem2}. With the same arguments, the third summand is also bounded by $C\beta^{\delta/(2+\delta)}_k$. For the second summand, we make use of the variation condition and the $P$-NED property
\begin{multline*}
	\left\|g_2(\bX''_i,\bX'_{i+k+2l})-g_2(\bX''_{i,l},\bX'_{i+k+2l,l})\right\|_2\\
	\leq 
	\big\|\{ g_2(\bX''_i,\bX'_{i+k+2l})-g_2(\bX''_{i,l},\bX'_{i+k+2l,l})\}
			\bEins_{\left\{\left|\bX''_i-\bX''_{i,l}\right|\leq\epsilon, \left|\bX'_{i+k+2l}-\bX'_{i+k+2l,l}\right|\leq\epsilon\right\}}\big\|_2 \\
	+ \big\|\{(g_2(\bX''_i,\bX'_{i+k+2l})-g_2(\bX''_{i,l},\bX'_{i+k+2l,l})\}
			\bEins_{\left\{\left|\bX''_i-\bX''_{i,l}\right|>\epsilon\text{ or } \left|\bX'_{i+k+2l}-\bX'_{i+k+2l,l}\right|>\epsilon\right\}}\big\|_2 \\
\leq \sqrt{L\epsilon}+2M^{2/(2+\delta)}2P\left(\left|\bX_{i+k+2l}-f_l(\bZ'_{i+k+l},\ldots,\bZ'_{i+k+3l})\right|>\epsilon\right)^{\frac{\delta}{2+\delta}} %\\
\leq C( \sqrt{\epsilon} +a_l^{\frac{\delta}{2+\delta}}\Phi^{\frac{\delta}{2+\delta}}(\epsilon)).
\end{multline*}
The proof is complete.
\end{proof}

%
%
%
%
%
%
%
%
\begin{lemma}\label{lem4} Under Assumptions \ref{ass3}, \ref{ass2} and \ref{ass1}, we have for $n_1<n_2\leq n$:
\begin{equation*}
\bigg\|\sum_{\substack{1\leq i<j\\n_1<j\leq n_2}}\big(g_2(\bX_i,\bX_j)-g_2(\bX_{i,l},\bX_{j,l})\big)\bigg\|_2\leq C(n_2-n_1)n^{1/4}.
\end{equation*}
\end{lemma}

\begin{proof}[Proof of Lemma \ref{lem4}] We set $l=\lfloor n^{1/4}\rfloor$ and $\epsilon=l^{-6}$. If $k<0$, we set $\beta_k=1.$ With Lemma \ref{lem3} and some straightforward calculus, we obtain
\begin{multline*}
	\left\|\sum_{\substack{1\leq i<j\\n_1<j\leq n_2}}\left(g_2(\bX_i,\bX_j)-g_2(\bX_{i,l},\bX_{j,l})\right)\right\|_2 
	\leq 
	C(n_2-n_1) \sum_{k=1}^n (\sqrt{\epsilon}+a_l^{\frac{\delta}{2+\delta}}\Phi^{\frac{\delta}{2+\delta}}(\epsilon)+\beta^{\frac{\delta}{2+\delta}}_{k-l})   \\
	\leq 
	C(n_2-n_1)\sum_{k=1}^n 
	\left( \lfloor n^{1/4}\rfloor^{-3} + a_l^{\frac{\delta}{2+\delta}}\Phi^{\frac{\delta}{2+\delta}}(\lfloor n^{1/4}\rfloor^{-6})+\beta^{\frac{\delta}{2+\delta}}_{k-\lfloor n^{1/4}\rfloor} \right)
\leq C(n_2-n_1)n^{1/4}. 
\end{multline*}
\end{proof}
%
%
%
%
%
%
%
%


As we approximate the random variables $\bX_i$ by $\bX_{i,l}=f_l(\bZ_{i-l},\ldots,\bZ_{i+l})$, we introduce the Hoeffding decomposition of the kernel 
$g$ with respect to these approximating random variables. Let $(\tilde{\bZ}_n)_{n\in\Z}$ be an independent copy of the sequence $(\bZ_n)_{n\in\Z}$ and $\tilde{\bX}_{i,l}=f_l(\tilde{\bZ}_{i-l},\ldots,\tilde{\bZ}_{i+l})$. We define
\[
U_l = Eg(\bX_{0,l},\tilde{\bX}_{0,l}), \quad 
g_{1,l}(x) = Eg(\bx,\tilde{\bX}_{0,l})-U_l, \quad 
g_{2,l}(\bx,\by) = g(\bx,\by)-g_{1,l}(\bx)-g_{1,l}(\by)-U_l.
\]

\begin{lemma}\label{lem4a} Under Assumptions \ref{ass3}, \ref{ass2} and \ref{ass1}, we have for $n_1<n_2\leq n$:
\begin{equation*}
	\bigg\|\sum_{\substack{1\leq i<j\\n_1<j\leq n_2}}\left(g_{2,l}(\bX_{i,l},\bX_{j,l})-g_2(\bX_{i,l},\bX_{j,l})\right)\bigg\|_2
	\leq 
	C(n_2-n_1)n^{1/4}.
\end{equation*}
\end{lemma}

\begin{proof}[Proof of Lemma \ref{lem4a}] Let $(\tilde{\bZ}_n)_{n\in\Z}$ be an independent copy of the sequence $(\bZ_n)_{n\in\Z}$ and $\tilde{\bX}_{i}=f_\infty((\tilde{\bZ}_{n+l})_{l\in\Z})$. Then
$g_2(\bx,\by) = g(\bx,\by)-Eg(\bx,\tilde\bX_j)-Eg(\tilde\bX_i,\by)+Eg(\tilde{\bX}_i,\bX_j)$ and
$g_{2,l}(\bx,\by)= g(\bx,\by)-Eg(\bx,\tilde{\bX}_{j,l})-Eg(\tilde{\bX}_{i,l},\by)+Eg(\tilde{\bX}_{i,l},\bX_{j,l})$.
for every $i, j, l \in \N$.
%\begin{align*}
	%g_2(\bx,\by)=&g(\bx,\by)-Eg(\bx,\tilde{\bX}_j)-Eg(\bX'_i,\by)+Eg(\tilde{\bX}_i,\bX_j)\\
	%g_{2,l}(\bx,\by)=&g(\bx,\by)-Eg(\bx,\tilde{\bX}_{j,l})-Eg(\tilde{\bX}_{i,l},\by)+Eg(\tilde{\bX}_{i,l},\bX_{j,l}).\\
%\end{align*}
So we can conclude that
\[ %begin{multline*}
	\left\|g_{2,l}(\bX_{i,l},\bX_{j,l})-g_2(\bX_{i,l},\bX_{j,l})\right\|_2
	\leq \left\|g(\bX_{i,l},\tilde{\bX}_{j,l})-g(\bX_{i,l},\tilde{\bX}_{j})\right\|_2
\]\[	
 \qquad	+ \left\|g(\tilde{\bX}_{i,l},\bX_{j,l})-g(\tilde{\bX}_{i},\bX_{j,l})\right\|_2
	+ \left\|g(\tilde{\bX}_{i,l},\bX_{j,l})- g(\tilde{\bX}_{i},\bX_j)\right\|_2 
\]
\[
	\leq C(\sqrt{\epsilon}+a_l^{\frac{\delta}{2+\delta}}\Phi^{\frac{\delta}{2+\delta}}(\epsilon)+\beta^{\frac{\delta}{2+\delta}}_k),
\]
where the bound in the last line can be proved along the lines of Lemma \ref{lem3}. The assertion of Lemma \ref{lem4a} then follows analogously to Lemma \ref{lem4}.
\end{proof}

%
%
%
%
%
%
%
%
%
\begin{lemma}\label{lem5}
Under Assumptions \ref{ass3}, \ref{ass2} and \ref{ass1}, we have for $n_1 < n_2\leq n$
\begin{equation*}
	E\left(\sum_{\substack{1\leq i<j\\n_1<j\leq n_2}}g_{2,l}(\bX_{i,l},\bX_{j,l})\right)^2\leq C(n_2-n_1)\,n\,l^2.
\end{equation*}
\end{lemma}
%
%
%
\begin{proof}[Proof of Lemma \ref{lem5}] By Lemma 1 of \citet{yoshihara:1976} we obtain
\begin{equation*}
		\left|E \{ g_{2,l}(\bX_{i_{(1)},l},\bX_{i_{(2)},l})g_{2,l}(\bX_{i_{(3)},l},\bX_{i_{(4)},l}) \} \right|\leq C\beta^{\delta/(2+\delta)}_{m-l}
\end{equation*}
with $m=\max\left\{i_{(2)}-i_{(1)},i_{(4)}-i_{(3)}\right\}$, where $i_{(1)}\leq i_{(2)} \leq i_{(3)}\leq i_{(4)}$ are the ordered indices $i_1,i_2,i_3,i_4$. 
To simplify the notation, we define $\beta_{m-l}=1$ for $m-l<0$. Note that, for given $m$, we have less than $n$ choices for $i_{(1)}$ and less than $n_2-n_1$ choices for $i_{(4)}$, which has to be either $i_2$ or $i_4$. If $m = i_{(2)}-i_{(1)}$, there are $m$ possibilities for $i_{(3)}$, and if $m=i_{(4)}-i_{(3)}$, there are $m$ possibilities for $i_{(2)}$. We conclude that
\begin{multline*}
	E\Bigg(\sum_{\substack{1\leq i<j\\n_1<j\leq n_2}}g_{2,l}(\bX_{i,l},\bX_{j,l})\Bigg)^2
	=	\sum_{\substack{1\leq i_1<j_1\\n_1<j\leq n_2}}\sum_{\substack{1\leq i<j\\n_1<j\leq n_2}}C\beta^{\delta/(2+\delta)}_{m-l}
	\ \leq \ C(n_2-n_1)n \sum_{m=1}^n m\beta^{\delta/(2+\delta)}_{m-l} \\
	\leq C(n_2-n_1)n\left(\sum_{m=1}^l m+l^2\sum_{m=l+1}^n (m-l) \beta^{\delta/(2+\delta)}_{m-l}\right)\leq C(n_2-n_1)nl^2.
\end{multline*}
The proof is complete.
\end{proof}

%
%
%
%
%
%
%
%
%
%
%
\begin{lemma}\label{lem6} Under Assumptions \ref{ass3}, \ref{ass2} and \ref{ass1}, we have
\begin{enumerate}[(i)]
\item
$ \displaystyle \Big\|\max_{n\leq 2^k}\sum_{1\leq i<j\leq n}g_2(\bX_i,\bX_j)\Big\|_2\leq C 2^{\frac{5}{4}k}k $ \ and
\item
$ \sum_{1\leq i<j\leq n}g_2(\bX_i,\bX_j)=O\left(n^{\frac{5}{4}}\log^2 (n)\right)$ almost surely.
\end{enumerate}
\end{lemma}
%
%
%
\begin{proof}[Proof of Lemma \ref{lem6} (i):] We will use Lemma \ref{lem4} and Lemma \ref{lem5} with $l=l_k=\lfloor 2^{\frac{1}{4}k}\rfloor$ and split the expectation into three parts:
\begin{multline*}
	\bigg\|\max_{n\leq 2^k}\Big|\sum_{1\leq i<j\leq n}g_2(\bX_i,\bX_j)\Big|\bigg\|_2 
	\leq \bigg\|\max_{n\leq 2^k}\Big|\sum_{1\leq i<j\leq n}\left(g_2(\bX_i,\bX_j)-g_2(\bX_{i,l},\bX_{j,l})\right)\Big|\bigg\|_2 \\
	+\bigg\|\max_{n\leq 2^k}\Big|\sum_{1\leq i<j\leq n}(g_{2,l}(\bX_{i,l},\bX_{j,l})-g_2(\bX_{i,l},\bX_{j,l}))\Big| \bigg\|_2\\
	+\bigg\|\max_{n\leq 2^k}\Big|\sum_{\substack{1\leq i<j\leq n}}g_{2,l}(\bX_{i,l},\bX_{j,l})\Big|\bigg\|_2=I_k+I\!\! I_k+I\!\! I\!\! I_k.
\end{multline*}
Now by Lemma \ref{lem4}
\begin{multline*}
	I_k  %= \bigg\|\max_{n\leq 2^k}\Big|\sum_{1\leq i<j\leq n}\left(g_2(\bX_i,\bX_j)-g_2(\bX_{i,l},\bX_{j,l})\right)\Big|\bigg\|_2\\
	\leq \bigg\|\sum_{j=1}^{2^k}\Big|\sum_{1\leq i<j}\left(g_2(\bX_i,\bX_j)-g_2(\bX_{i,l},\bX_{j,l})\right)\Big|\bigg\|_2 \\
	\leq \sum_{j=1}^{2^k}\bigg\|\sum_{1\leq i<j}\left(g_2(\bX_i,\bX_j)-g_2(\bX_{i,l},\bX_{j,l})\right)\bigg\|_2
		\leq C 2^{\frac{5}{4}k}.
\end{multline*}
Similarly, we get by Lemma \ref{lem4a} that $I\!\! I_k\leq C 2^{\frac{5}{4}}$. To deal with $I\!\! I\!\! I_k$, we define the random variables
$Y_{j,l}=\sum_{1\leq i<j}g_{2,l}(\bX_{i,l},\bX_{j,l})$
and rewrite $I\!\! I\!\! I_k$ as $\big\|\max_{n\leq 2^k}|\sum_{j=1}^n Y_{j,l}|\big\|_2$.
%
%
As we have $E(\sum_{j=n_1}^{n_2} Y_{j,l})^2\leq C(n_2-n_1)n^{3/2}$ by Lemma \ref{lem5}, we can use Theorem 1 of \citet{moricz1976moment} to obtain
$ E(\max_{n\leq 2^k}\sum_{j=1}^n Y_{j,l})^2\leq C2^{5k/2}k^2$, which completes the proof of part (1). 

\emph{Part (ii):} It suffices to prove that
\begin{equation*}
\max_{n\leq 2^k}\sum_{1\leq i<j\leq n}g_2(\bX_i,\bX_j)=O\left( 2^{\frac{5}{4}k}k^2\right)
\end{equation*}
almost surely. By means of the Chebychev inequality, we have for any $\epsilon>0$
\begin{multline*}
\sum_{k=1}^\infty P\left(\max_{n\leq 2^k}\sum_{1\leq i<j\leq n}g_2(\bX_i,\bX_j)>\epsilon2^{5k/4}k^2 \right)\\
\leq \sum_{k=1}^\infty\frac{1}{2^{5k/2}k^4}E\left(\max_{n\leq 2^k}\sum_{1\leq i<j\leq n}g_2(\bX_i,\bX_j)\right)^2\leq C\sum_{k=1}^\infty  \frac{2^{5k/2}k^2}{2^{5k/2}k^4}=C\sum_{k=1}^\infty \frac{1}{k^2}<\infty,
\end{multline*}
and the almost sure convergence follows by the Borel--Cantelli lemma.
\end{proof}
%
%




%%%%--------------------------------------------------------------------------------------------------------------------------------------------------
%%%===================================================================================================================================================
% das war in Appendix C

%%%%--------------------------------------------------------------------------------------------------------------------------------------------------
%%%===================================================================================================================================================





Towards the proof of Theorem \ref{theo3}, we further state and prove Lemmas \ref{lem9} to \ref{lem11}.
%
%
%
\begin{lemma}\label{lem9} Under Assumptions \ref{ass3}, \ref{ass2} and \ref{ass1}, we have for any constants $(c_i)_{i\in\N}$
\begin{equation*}
E\bigg(\sum_{i=1}^ng_1(\bX_i)c_i\bigg)^2\leq Cn\left(\max_{i=1,\ldots,n}|c_i|\right)^2,
\end{equation*}
\end{lemma}
%
%
\begin{proof}[Proof of Lemma \ref{lem9}] By Assumption \ref{ass1} and Lemma \ref{lem:neu1}, the process $(g_1(\bX_n))_{n\in\Z}$ is $L_2$-NED with approximation constants $a_{l,2}=O(l^{-3})$. We have the following bound for the autocovariance: 
\[
	\left|Eg_1(\bX_i)g_1(\bX_{i+k})\right|
\]\[
	\leq	
	\left|E \{E(g_1(\bX_i)|\F_{i-l}^{i+l})E(g_1(\bX_{i+k})|\F_{i+k-l}^{i+k+l})\}\right|
	+ 
	\left|E\{g_1(\bX_i)\left(g_1(\bX_{i+k})-E(g_1(\bX_{i+k})|\F_{i+k-l}^{i+k+l})\right)\}\right|
\]\[
	\quad +\left|E\left(E(g_1(\bX_{i+k})|\F_{i+k-l}^{i+k+l})\left(g_1(\bX_{i})-E(g_1(\bX_{i})|\F_{i-l}^{i+l})\right)\right)\right|
\]\[
	\leq 10\left\|E(g_1(\bX_{i+k})|\F_{i+k-l}^{i+k+l})\right\|_{2+\delta}^2\beta^{\frac{\delta}{2+\delta}}_{k-2l}+2\left\|g_1(\bX_i)\right\|_2	
	\left\|g_1(\bX_{i+k})-E(g_1(\bX_{i+k})|\F_{i+k-l}^{i+k+l})\right\|_2
\]\[
	\leq C(a_{l,2}+\beta^{\frac{\delta}{2+\delta}}_{l}),
\]
where we used the inequality by \citet{davydov1970invariance} and set $l=\lfloor\frac{k}{3}\rfloor$. Recall that $\F_i^j$ is defined as the $\sigma$-algebra generated by $\bZ_i,\ldots,\bZ_j$. Now it follows from the stationarity of the process that
\begin{multline*}
E\left(\sum_{i=1}^ng_1(\bX_i)c_i\right)^2\leq \sum_{i,j=1}^n\left|Eg_1(\bX_i)g_1(\bX_{j})\right|\left|c_i\right|\left|c_j\right|
\\\leq \left(\max_{i=1,\ldots,n}|c_i|\right)^2\sum_{i,j=1}^n\left|Eg_1(\bX_i)g_1(\bX_{j})\right|\leq C\left(\max_{i=1,\ldots,n}|c_i|\right)^2n\sum_{k=0}^n\left|Eg_1(\bX_0)g_1(\bX_{k})\right|\\
\leq C\left(\max_{i=1,\ldots,n}|c_i|\right)^2n\sum_{l=0}^{\lfloor\frac{n}{3}\rfloor}(a_{l,2}+\beta^{\frac{\delta}{2+\delta}}_{l})\leq Cn\left(\max_{i=1,\ldots,n}|c_i|\right)^2,
\end{multline*}
and the Lemma is proved.
\end{proof}

%
%
%
%
%
%
%
%
%
%
%
%
\begin{lemma}\label{lem10} Under the Assumptions \ref{ass3}, \ref{ass2} and \ref{ass1} for any constants $(c_{ij})_{i,j \in\N}$
\begin{equation*}
\left|E\sum_{j_1,j_2,j_3,j_4=1}^ng_2(\bX_{j_1},\bX_{j_2})g_2(\bX_{j_3},\bX_{j_4})c_{j_1j_3}\right|\leq C\max_{i,j\in\{1,\ldots,n\}}\left|c_{ij}\right|n^{5/2}.
\end{equation*}
\end{lemma}

\begin{proof}[Proof of Lemma \ref{lem10}] Recall that we abbreviate $f_l(\bZ_{j-l},\ldots,\bZ_{j+l})$ by $\bX_{j,l}$. We use the triangle inequality to obtain
\[
\bigg|E\sum_{j_1,j_2,j_3,j_4=1}^ng_2(\bX_{j_1},\bX_{j_2})g_2(\bX_{j_3},\bX_{j_4})c_{j_1j_3}\bigg|\] \[
\leq \bigg|E\sum_{j_1,j_2,j_3,j_4=1}^ng_{2,l}({\bX}_{j_1,l},{\bX}_{j_2,l})g_{2,l}({\bX}_{j_3,l},{\bX}_{j_4,l})c_{j_1j_3}\bigg|\] \[
+\bigg|E\sum_{j_1,j_2,j_3,j_4=1}^n\left(g_{2,l}({\bX}_{j_1,l},{\bX}_{j_2,l})-g_{2}({\bX}_{j_1,l},{\bX}_{j_2,l})\right)\left(g_{2,l}({\bX}_{j_3,l},{\bX}_{j_4,l})-g_{2}({\bX}_{j_3,l},{\bX}_{j_4,l})\right)c_{j_1j_3}\bigg|\] \[
+\bigg|E\sum_{j_1,j_2,j_3,j_4=1}^n\left(g_{2}({\bX}_{j_1,l},{\bX}_{j_2,l})-g_{2}(\bX_{j_1},\bX_{j_2})\right)\left(g_{2}({\bX}_{j_3,l},{\bX}_{j_4,l})-g_{2}(\bX_{j_3},\bX_{j_4})\right)c_{j_1j_3}\bigg|\] \[
+\bigg|E\sum_{j_1,j_2,j_3,j_4=1}^ng_{2,l}({\bX}_{j_1,l},{\bX}_{j_2,l})\left(g_{2,l}({\bX}_{j_3,l},{\bX}_{j_4,l})-g_{2}({\bX}_{j_3,l},{\bX}_{j_4,l})\right)c_{j_1j_3}\bigg|\displaybreak[0]\] \[
+\bigg|E\sum_{j_1,j_2,j_3,j_4=1}^n\left(g_{2,l}({\bX}_{j_1,l},{\bX}_{j_2,l})-g_{2}({\bX}_{j_1,l},{\bX}_{j_2,l})\right)g_{2,l}({\bX}_{j_3,l},{\bX}_{j_4,l})c_{j_1j_3}\bigg|\] \[
+\bigg|E\sum_{j_1,j_2,j_3,j_4=1}^ng_{2,l}({\bX}_{j_1,l},{\bX}_{j_2,l}))\left(g_{2}({\bX}_{j_3,l},{\bX}_{j_4,l})-g_{2}(\bX_{j_3},\bX_{j_4})\right)c_{j_1j_3}\bigg|\displaybreak[0]\] \[
+\bigg|E\sum_{j_1,j_2,j_3,j_4=1}^n\left(g_{2}({\bX}_{j_1,l},{\bX}_{j_2,l})-g_{2}(\bX_{j_1},\bX_{j_2})\right)g_{2,l}({\bX}_{j_3,l},{\bX}_{j_4,l})c_{j_1j_3}\bigg|\] \[
+\bigg|E\sum_{j_1,j_2,j_3,j_4=1}^n\left(g_{2,l}({\bX}_{j_1l},{\bX}_{j_2,l})-g_{2}({\bX}_{j_1,l},{\bX}_{j_2,l})\right)\left(g_{2}({\bX}_{j_3,l},{\bX}_{j_4,l})-g_{2}(\bX_{j_3},\bX_{j_4})\right)c_{j_1j_3}\bigg|\] \[
+\bigg|E\sum_{j_1,j_2,j_3,j_4=1}^n\left(g_{2}({\bX}_{j_1,l},{\bX}_{j_2,l})-g_{2}(\bX_{j_1},\bX_{j_2})\right)\left(g_{2,l}({\bX}_{j_3,l},{\bX}_{j_4,l})-g_{2}({\bX}_{j_3,l},{\bX}_{j_4,l})\right)c_{j_1j_3}\bigg|\] \[
=I_n+I\!\! I_n+I\!\! I\!\! I_n+I\! V_n+V_n+V\!\! I_n+V\!\! I\!\! I_n+V\!\! I\!\! I\!\! I_n+I\!\! X_n.
\]
% 
%
%
We will establish the bound only for some of the summands in order to keep this proof short. The bound for $I_n$ can be shown in the same way as Lemma \ref{lem5}. For $I\!\! I_n$, $I\!\! I\!\! I_n$, $V\!\! I\!\! I\!\! I_n$ and $I\!\! X_n$, we use the H\"older inequality and can then proceed similar to the proof of Lemma \ref{lem4} and \ref{lem4a}. For example by Lemmas \ref{lem3} and \ref{lem4a}, we have
\[
	V\!\! I\!\! I\!\! I_n
	\leq 
	\sum_{j_1,j_2,j_3,j_4=1}^n
	\left\|g_{2,l}({\bX}_{j_1,l},{\bX}_{j_2,l})-g_{2}({\bX}_{j_1,l},{\bX}_{j_2,l})\right\|_2
	\left\|g_{2}({\bX}_{j_3,l},{\bX}_{j_4,l})-g_{2}(\bX_{j_3},\bX_{j_4})\right\|_2c_{j_1j_3}
\]\[	
	\leq 
	C\max_{i,j\in\{1,\ldots,n\}}\left|c_{ij}\right|
	\sum_{j_1,j_2,j_3,j_4=1}^n
	(\sqrt{\epsilon}+a_l^{\frac{\delta}{2+\delta}}\Phi^{\frac{\delta}{2+\delta}}(\epsilon)+\beta^{\frac{\delta}{2+\delta}}_{|j_2-j_1|})
	(\sqrt{\epsilon}+a_l^{\frac{\delta}{2+\delta}}\Phi^{\frac{\delta}{2+\delta}}(\epsilon)+\beta^{\frac{\delta}{2+\delta}}_{|j_4-j_3|})
\]\[
	\leq C\max_{i,j\in\{1,\ldots,n\}}\left|c_{ij}\right|n^{5/2}
\] 
with $l=\lfloor n^{1/4}\rfloor$ and $\epsilon=l^{-6}$. For the summand $I\! V_n$, the H\"older inequality is used in a slightly different way
\[
	I\! V_n\leq \sum_{j_3,j_4=1}^n
	\bigg|
		E 
		\bigg\{ 
			\sum_{j_1,j_2=1}^ng_{2,l}({\bX}_{j_1,l},{\bX}_{j_2,l})
			\left(g_{2,l}({\bX}_{j_3,l},{\bX}_{j_4,l})-g_{2}({\bX}_{j_3,l},{\bX}_{j_4,l})\right)c_{j_1j_3} 
		\bigg\} 
	\bigg|
\]\[
	\leq \sum_{j_3,j_4=1}^n
	\bigg\|\sum_{j_1,j_2=1}^ng_{2,l}({\bX}_{j_1,l},{\bX}_{j_2,l})c_{j_1j_3}\bigg\|_2
	\big\|g_{2,l}({\bX}_{j_3,l},{\bX}_{j_4,l})-g_{2}({\bX}_{j_3,l},{\bX}_{j_4,l})\big\|_2
\]\[
	\leq 
	C\sum_{j_3,j_4=1}^n n^{5/4}\max_{i,j\in\{1,\ldots,n\}}\left|c_{ij}\right|(\sqrt{\epsilon}+a_l^{\frac{\delta}{2+\delta}}\Phi^{\frac{\delta}{2+\delta}}(\epsilon)+\beta^{\frac{\delta}{2+\delta}}_{|j_4-j_3|})\leq C\max_{i,j\in\{1,\ldots,n\}}\left|c_{ij}\right|n^{5/2}.
\]
A similar treatment of the summands $V_n$, $V\!\! I_n$ and $V\!\! I\!\! I_n$ completes the proof.
\end{proof}



%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
\begin{lemma}\label{lem11} Under Assumptions \ref{ass3}, \ref{ass2}, \ref{ass1} and \ref{ass4}, we have, for $n \to \infty$,
\begin{equation*}
	E
	\bigg|
		\sum_{r=-n}^n\frac{1}{n}\sum_{i=1}^{n-|r|}\left(g_1(\bX_i)g_1(\bX_{i+|r|})-\hat{g}_1(\bX_i)\hat{g}_1(\bX_{i+|r|})\right)\kappa(|r|/b_n)
	\bigg|
	\rightarrow 0.
\end{equation*}
\end{lemma}

\begin{proof}[Proof of Lemma \ref{lem11}] We first expand the difference of $g_1$ and its estimator $\hat{g}_1$ as
\begin{equation*}
	g_1(\bx)-\hat{g}_1(\bx)=\frac{1}{n}\sum_{i=1}^ng_1(\bX_i)-\frac{1}{n}\sum_{i=1}^ng_2(\bx,\bX_i)+\frac{1}{n^2}\sum_{i,j=1}^ng_2(\bX_i,\bX_j).
\end{equation*}
With the help of this, we split the expectation into six parts and apply the triangle inequality:
\[ \textstyle
	E\Big|\sum_{r=-n}^n\frac{1}{n}\sum_{i=1}^{n-|r|}\left(g_1(\bX_i)g_1(\bX_{i+|r|})-\hat{g}_1(\bX_i)\hat{g}_1(\bX_{i+|r|})\right)\kappa(|r|/b_n)\Big|\]\[ \textstyle
	\leq E\Big|\sum_{r=-n}^n\frac{1}{n}\sum_{i=1}^{n-|r|}\left(g_1(\bX_i)-\hat{g}_1(\bX_i)\right)g_1(\bX_{i+|r|})\kappa(|r|/b_n)\Big|
\]\[ \textstyle
	+ E\Big|\sum_{r=-n}^n\frac{1}{n}\sum_{i=1}^{n-|r|}\left(g_1(\bX_{i+|r|})-\hat{g}_1(\bX_{i+|r|})\right)\hat{g}_1(\bX_{i})\kappa(|r|/b_n)\Big|
\]\[ \textstyle
	\leq E\Big|\sum_{r=-n}^n\frac{1}{n}\sum_{i=1}^{n-|r|}\frac{1}{n}\sum_{j=1}^ng_1(\bX_j)g_1(\bX_{i+|r|})\kappa(|r|/b_n)\Big|\displaybreak[0]
\]\[ \textstyle
	+ E\Big|\sum_{r=-n}^n\frac{1}{n}\sum_{i=1}^{n-|r|}\frac{1}{n}\sum_{j=1}^ng_2(\bX_i,\bX_j)g_1(\bX_{i+|r|})\kappa(|r|/b_n)\Big|\displaybreak[0]
\]\[ \textstyle
	+ E\Big|\sum_{r=-n}^n\frac{1}{n}\sum_{i=1}^{n-|r|}\frac{1}{n^2}\sum_{j_1,j_2=1}^ng_2(\bX_{j_1},\bX_{j_2})g_1(\bX_{i+|r|})\kappa(|r|/b_n)\Big|
\]\[ \textstyle
	+ E\Big|\sum_{r=-n}^n\frac{1}{n}\sum_{i=1}^{n-|r|}\frac{1}{n}\sum_{j=1}^ng_1(\bX_j)\hat{g}_1(\bX_{i})\kappa(|r|/b_n)\Big|
\]\[ \textstyle
	+ E\Big|\sum_{r=-n}^n\frac{1}{n}\sum_{i=1}^{n-|r|}\frac{1}{n}\sum_{j=1}^ng_2(\bX_{i+|r|},\bX_j)\hat{g}_1(\bX_{i})\kappa(|r|/b_n)\Big|
\]\[ \textstyle
	+ E\Big|\sum_{r=-n}^n\frac{1}{n}\sum_{i=1}^{n-|r|}\frac{1}{n^2}\sum_{j_1,j_2=1}^ng_2(\bX_{j_1},\bX_{j_2})\hat{g}_1(\bX_{i})\kappa(|r|/b_n)\Big|
\] \[
	\qquad = I_n+I\!\! I_n+I\!\! I\!\! I_n+I\! V_n+V_n+V\!\! I_n.
\]
%
%
%
For the first summand $I_n$, we use the H\"older inequality and Lemma \ref{lem9} with constants $c_i=\sum_{i_2=1}^n\kappa(|i-i_2|/b_n)=O(b_n)$ and obtain
\[
	\textstyle
	I_n = 
	E\Big|\frac{1}{n}\sum_{j=1}^ng_1(\bX_j)\Big|\cdot\Big|\frac{1}{n}\sum_{i=1}^{n}g_1(\bX_{i})\sum_{i_2=1}^n\kappa(|i-i_2|/b_n)\Big|
\]\[
	\qquad \leq 
	\textstyle
	\Big\|\frac{1}{n}\sum_{j=1}^ng_1(\bX_j)\Big\|_2
	\Big\|\frac{1}{n}\sum_{i=1}^{n}g_1(\bX_{i})\sum_{i_2=1}^n\kappa(|i-i_2|/b_n)\Big\|_2\leq C\frac{1}{\sqrt{n}}\frac{1}{\sqrt{n}}b_n\rightarrow 0
\] %end{multline*}
as $n\rightarrow\infty$ due to the assumptions on $b_n$. %
%
%
%
%
%
For $I\!\! I_n$, we use Lemma \ref{lem10} to obtain
\[ %begin{multline*}
	\textstyle
	I\!\! I_n = E\bigg|\frac{1}{n}\sum_{i_1=1}^n\frac{1}{n}\sum_{i,j=1}^{n}g_2(\bX_i,\bX_j)g_1(\bX_{i_1})\kappa(|i-i_1|/b_n)\bigg|
\]\[
	\textstyle
	\leq 
	E
	\left[
		\big\{ \frac{1}{n}\sum_{i=1}^n g_1(\bX_{i_1})^2 \big\}^{1/2} 
		\Big\{
			\frac{1}{n}\sum_{i_1=1}^n \big(\frac{1}{n}\sum_{i,j=1}^{n}g_2(\bX_i,\bX_j)\kappa(|i-i_1|/b_n)\big)^2
		\Big\}^{1/2}
	\right]
\]\[
	\textstyle
	\leq \Big[ E \big\{  \frac{1}{n}\sum_{i=1}^n g_1(\bX_{i_1})^2  \big\} \Big]^{1/2}
\]\[
	\qquad \qquad
	\bigg[ E \big\{ 
		\frac{1}{n^3}\sum_{j_1,j_2,j_3,j_4=1}^ng_2(\bX_{j_1},\bX_{j_2})g_2(\bX_{j_3},\bX_{j_4})\sum_{i_1=1}^n\kappa(|j_1-i_1|/b_n)\kappa(|j_3-i_1|/b_n)
	\big\} \bigg]^{1/2}
\]\[
	\leq C\sqrt{\frac{n^{5/2}b_n}{n^3}}\rightarrow 0,
\]
since $c_{j_1j_2}=\sum_{i_1=1}^n\kappa(|j_1-i_1|/b_n)\kappa(|j_3-i_1|/b_n)=O(b_n)=o(\sqrt{n})$ and 
$	E\{\frac{1}{n}\sum_{i=1}^n g_1(\bX_{i_1})^2\}^2 \leq E\{ g_1(\bX_{0})^2 \}^2<\infty$
due to Assumption \ref{ass2}. For the third summand $I\!\! I\!\! I_n$, we use again the H\"older inequality and Lemma \ref{lem9} to get
\[	
	\textstyle
	I\!\! I\!\! I_n = 
	E\Big|\sum_{r=-n}^n\frac{1}{n}\sum_{i=1}^{n-|r|}\frac{1}{n^2}\sum_{j_1,j_2=1}^ng_2(\bX_{j_1},\bX_{j_2})g_1(\bX_{i+|r|})\kappa(|r|/b_n)\Big|
\]\[
	\leq 
	\bigg\| \frac{1}{n^2}\sum_{j_1,j_2=1}^ng_2(\bX_{j_1},\bX_{j_2})\bigg\|_2
	\bigg\|\frac{1}{n}\sum_{i=1}^{n-|r|}g_1(\bX_{i+|r|})\sum_{i_1=1}^n\kappa(|i-i_1|/b_n)\bigg\|_2
	\leq C\frac{1}{n^{3/4}}\frac{1}{\sqrt{n}}b_n
	\rightarrow 0.
\]
The convergence of the remaining parts $I\! V_n$, $V_n$ and $V\!\! I_n$ can be shown in the same way.
\end{proof}
%
%
%
%
%
%
%
%
%
%

%\bibliographystyle{chicago}
%\nocite{*}
\small
%\bibliography{KendallsTau-References}

\begin{thebibliography}{}

\bibitem[\protect\citeauthoryear{Borovkova, Burton, and Dehling}{Borovkova
  et~al.}{2001}]{borovkova:burton:dehling:2001}
Borovkova, S., R.M. Burton, and H.~Dehling (2001) 
\newblock Limit theorems for functionals of mixing processes with applications
  to {$U$}-statistics and dimension estimation.
\newblock {\em Transactions of the American Mathematical Society\/}~{\em
  353}, 4261--4318.

\bibitem[\protect\citeauthoryear{Davydov}{Davydov}{1970}]{davydov1970invariance}
Davydov, Y.A. (1970) 
\newblock The invariance principle for stationary processes.
\newblock {\em Theory of Probability and its Applications\/}~{\em 15},
  487--498.

\bibitem[\protect\citeauthoryear{M{\'o}ricz}{M{\'o}ricz}{1976}]{moricz1976moment}
M{\'o}ricz, F. (1976) 
\newblock Moment inequalities and the strong laws of large numbers.
\newblock {\em Probability Theory and Related Fields\/}~{\em 35},
  299--314.

\bibitem[\protect\citeauthoryear{Yoshihara}{Yoshihara}{1976}]{yoshihara:1976}
Yoshihara, K.-I. (1976) 
\newblock Limiting behavior of {$U$}-statistics for stationary, absolutely
  regular processes.
\newblock {\em Zeitsschrift f{\"u}r Wahrscheinlichkeitstheorie und Verwandte
  Gebiete\/}~{\em 35}, 237--252.

\end{thebibliography}


\end{document}
