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\begin{document}




\newpage



\title{ Online Supplement to ``Goodness-of-fit tests for multivariate copula-based time series models''}

\author{Betina Berghaus and Axel B\"ucher\footnote{Ruhr-Universit\"at Bochum,
Fakult\"at f\"ur Mathematik, Universit\"atsstr.\ 150, 44780 Bochum, Germany. 
{E-mail:} betina.berghaus@rub.de, axel.buecher@rub.de.
}
}


\maketitle

\begin{abstract}
This supplementary material contains a detailed description of a procedure to simulate a dependent multiplier sequence,  a section illustrating the runtime performance of the proposed tests and  a section containing the proofs of Lemma~\ref{lem:boot:hat}, \ref{le:08} and \ref{lem:taylor} from the main paper.
\end{abstract}


\appendix
\setcounter{section}{4}

\section{Practical issues for the choice of the multipliers} \label{app:mult}
\def\theequation{E.\arabic{equation}}
\setcounter{equation}{0}

As described in Section~\ref{subsec:mult}, the bootstrap version of the test statistic in Section~\ref{sec:test} depends crucially on a sequence of dependent multipliers satisfying the conditions in Definition~\ref{def:mult} in Appendix~\ref{app:cond}. Within this section, we present a slight modification of the approach in \cite{BucKoj14}, Section~5.2, originating from \cite{Buh93} and \cite{PolWhi04}, showing how to simulate such a sequence. 



\subsection{Simulating dependent multipliers} \label{subsec:mult2}
First of all, choose a bandwidth parameter $\ell_n$ (see Section \ref{subsec:elln} below) and set $b_n = \textrm{round} \{(\ell_n+1)/2 \}$. Let $\kappa$ be a positive, bounded function which is symmetric around zero such that $\kappa (x) >0$ for all $x \in (-1,1)$. Within the simulation study, we use the Parzen-kernel defined as
\[ 
\kappa_P(x) = (1-6x^2+6 \vert x \vert^3 ) \ind (\vert x  \vert \leq 1/2) + 2(1-\vert x \vert)^3  \ind(1/2 < \vert x \vert \leq1).
\]
 Now, for $j =1, \dots , \ell_n$, define weights $\tilde \omega _{jn} = \omega_{jn} (\sum_{i=1}^{\ell_n} \omega_{in}^2)^{-1/2}$ where $\omega_{jn}= \kappa \{ (j- b _n )/b_n \}$.
Finally, let $\xi_1, \dots , \xi_{n+2 b_n - 2}$ be i.i.d.\ random variables which are independent from the sample $\vect X_1, \dots, \vect X_n$ with $\mathbb{E}[\xi_1]=0$, $\mathbb{E}[\xi_1^2]=1$ and $\mathbb{E}[\vert \xi_1 \vert^m] < \infty$, for any $m \geq 1$. Then, the sequence of random variables $Z_{1,n}, \dots ,Z_{n,n}$ defined as
\[ 
Z_{i,n} = \sum_{j=1}^{\ell_n} \tilde \omega_{jn}\xi_{j+i-1}
\]
asymptotically satisfies the conditions in Definition~\ref{def:mult}, see \cite{BucKoj14}. For a detailed discussion on the properties of these multipliers and an alternative simulation procedure for the multpliers see \cite{BucKoj14}.



\subsection{Choosing the bandwidth parameter $\ell_n$}\label{subsec:elln}

The simulation method described in the previous section depends on the choice of the `bandwidth' parameter $\ell_n$. 
Within the context of empirical copulas, \cite{BucKoj14} derived a closed-form formula for a theoretically optimal choice, where optimality is to be understood as optimality  with respect to a certain MSE-minimizing criterion. Moreover, they proposed a data-adaptive estimation procedure for this (theoretically) optimal bandwidth. 
In the following, we adapt their approach to the processes underlying the goodness-of-fit tests in Section~\ref{sec:test}.

For $ \vect u , \vect v \in [0,1]^d$, let $\sigma( \vect u , \vect v ) $ denote the characterizing covariance kernel of the process $\vect u \mapsto \mathbb{C}_C(\vect u) -\nabla C_{\theta_0}(\vect u) \Theta$. Its bootstrap approximation, for a fixed $b\in\{1, \dots, B\}$, is given by $\Cb_n^{(b)} (\vect u) - \nabla C_{\theta_n}(\vect u)  \Theta_n^{(b)}$. The main idea of the subsequent developments is as follows: if $\hat \sigma_n$ denotes the covariance kernel of the bootstrap process, conditional on the data, then a theoretically optimal choice of $\ell_n$ is given by the minimizer of the integrated mean squared error of $\hat \sigma_n(\vect u, \vect v)$, seen as an estimator for $\sigma(\vect u, \vect v)$, with respect to $\ell_n$. Unfortunately, necessary closed form expressions for the mean or the variance of $\hat \sigma_n$ are out of reach, whence we follow the proposal of \cite{BucKoj14} and consider an asymptotically equivalent form of $\Cb_n^{(b)} (\vect u) - \nabla C_{\theta_n}(\vect u)  \Theta_n^{(b)}$ instead, for which calculations are feasible.

More precisely,  define
$
\tilde \Bb_n^{(b)}(\vect u)   = n^{-1/2} \sum_{i=1}^n  Z_{i,n}^{(b)}\{\ind( \vect U_i \leq \vect u ) -C_{\theta_0}(\vect u)\}$ and  
$\tilde \Theta_n^{(b)} = n^{-1/2}\sum_{i=1}^n  Z_{i,n}^{(b)} \{ J_{\theta_0}(\vect U_i) + K_{i,\theta_0} \}$, 
where $K_{i,\theta_0}$ is defined in Theorem~\ref{theo:main},
and let $\tilde{ \mathbb{C}}_n^{(b)} (\vect u) = \tilde{\mathbb{B}}_n^{(b)}(\vect u ) - \sum_{\dimi=1}^d C_{\theta_0}^{[\dimi]}(\vect u ) \tilde{\mathbb{B}}_n^{(b)}(\vect u^{\dimi})$. It follows from the arguments in the proof of Theorem~\ref{theo:main} that
\[
 \sup_{\vect u \in [0,1]^d} \left|  \Cb_n^{(b)} (\vect u) - \nabla C_{\theta_n}(\vect u) \Theta_n^{(b)} -  \{\tilde \Cb_n^{(b)} (\vect u) - \nabla C_{\theta_0}(\vect u) \tilde \Theta_n^{(b)} \} \right| =o_P(1).
\]
In contrast to $\hat \sigma_n$, the (conditional) covariance kernel of the (unobservable) process $\tilde \Cb_n^{(b)} (\vect u) - \nabla C_{\theta_0}(\vect u) \tilde \Theta_n^{(b)}$ can be calculated explicitly, at least up to the first-order terms. We have
\begin{align*}
\tilde \sigma_n(\vect u, \vect v) 
& = 
\Cov_{Z} \{ \tilde{ \mathbb{C}}_n^{(b)}(\vect u) -\nabla C_{\theta_0 }(\vect u) \tilde{ \Theta}^{(b)},\tilde{\mathbb{C}}_n^{(b)}(\vect v) -\nabla C_{\theta_0 }(\vect v) \tilde{ \Theta}^{(b)} \} \\
&= \frac{1}{n} \sum_{i,j=1}^n \mathbb{E}_{Z}[Z_{i,n}^{(b)}Z_{j,n}^{(b)}] f(\vect U_i,\vect u) f(\vect U_j,\vect v) \\
&= \frac{1}{n} \sum_{i,j=1}^n \dmd \{ (i-j)/\ell_n\} f(\vect U_i, \vect u) f(\vect U_j,\vect v)
\end{align*}
where $\Cov_Z$ and $\mathbb{E}_Z$ denote covariance and expectation conditional on the data, respectively, and where, for $i = 1 , \dots , n$ and $\vect u \in [0,1]^d$,
\begin{multline*}
f(\vect U_i , \vect u) = \ind (\vect U_i \leq \vect u) - C_{\theta_0} (\vect u) - \sum_{\dimi=1}^d  C_{\theta_0}^{[\dimi]}(\vect u) \{ \ind (U_{i \dimi} \leq u_\dimi) -u_\dimi \} \\
-\nabla C_{\theta_0}(\vect u) \{ J_{\theta_0}(\vect U_i ) + K_{\theta_0 , i}\}.
\end{multline*}
Mimicking the proofs of Proposition 5.1 and Proposition 5.2 in \cite{BucKoj14}, we obtain the following results regarding bias and variance of $\tilde \sigma_n$, seen as an estimator for $\sigma$.  


\begin{lemma} \label{lem:opt}
Additionally to the conditions assumed in (iii) of Theorem~\ref{theo:main} suppose that $a$ defined in Condition~\ref{cond:mixBet} satisfies $a>3(2+\nu)/\nu$ and that $\dmd$ defined in Definition \ref{def:mult} is  twice continuously differentiable on $[-1,1] $ with $\dmd''(0)\neq 0$. Then, for any $\vect u , \vect v \in [0,1]^d$, 
\begin{align*}
\mathbb{E}[\tilde \sigma_n (\vect u , \vect v ) ] - \sigma (\vect u, \vect v) = \frac{\Gamma(\vect u, \vect v)}{\ell_n^2} + r_{n,1}(\vect u, \vect v),
\end{align*}
where $\sup_{\vect u , \vect v \in [0,1]^d} \vert r_{1,n}(\vect u , \vect v) \vert =o(\ell_n^{-2})$ and $\Gamma(\vect u , \vect v) = \frac{\dmd''(0)}{2} \sum_{k \in \mathbb{Z}} k^2 \gamma(k, \vect u , \vect v)$ with $\gamma(k,\vect  u , \vect v) = \Cov\{ f(\vect U_0,\vect u ), f(\vect U_k , \vect v ) \})$.

Additionally, provided the function
$\dmd$ is Lipschitz-continuous 
and provided
$
\int_{[0,1]^d } \prod_{\dimi=1}^d r_\dimi(u_\dimi)^{4+2 \nu } \mathrm{d}C_{\theta_0}(\vect u) < \infty,
$
where $r_1, \dots, r_d$ are defined in Condition \ref{klasse2}, then 
\begin{align*}
\Var \{ \tilde  \sigma_n(\vect u, \vect v) \} = \frac{\ell_n}{n} \Delta(\vect u , \vect v) + r_{n,2} (\vect u ,\vect v),
\end{align*}
where  $\Delta (\vect  u, \vect v )=\int_{-1}^1 \dmd(x)^2 \mathrm{d}x \{ \sigma(\vect u , \vect u) \sigma (\vect v ,\vect v) +\sigma(\vect u , \vect v)^2 \} $ and where the remainder term satisfies $\sup_{\vect u , \vect v \in [0,1]^d} \vert r_{n,2}(\vect u , \vect v) \vert = o (\ell_n / n)$.
\end{lemma}


As a consequence of Lemma~\ref{lem:opt}, the (pointwise) mean integrated squared error of $\tilde \sigma_n (\vect u, \vect v) $ can be written as
 \begin{align*}
\mathrm{MSE}\{ \tilde \sigma_n(\vect u ,\vect  v ) \}  = \frac{\{ \Gamma (\vect u , \vect v)\}^2}{\ell_n^4} + \Delta(\vect u ,  \vect v) \frac{\ell_n}{ n }  + r_{n,1}^2( \vect u, \vect v) + r_{n,2}(\vect u ,\vect v).
\end{align*}
Furthermore, the integrated mean squared error is given by
\begin{align*}
\mathrm{IMSE}(\tilde \sigma_n)= 
\int_{[0,1]^{2d}} \mathrm{MSE}\{ \tilde \sigma_n(\vect u ,\vect  v ) \} \mathrm{d}(\vect u, \vect v  )
=\frac{\bar \Gamma^2}{\ell_n^4} + \bar \Delta \frac{\ell_n}{n}+ o( \ell_n^{-4})+o(\ell_n / n),
\end{align*}
where $\bar \Gamma = \int_{[0,1]^{2d}}\Gamma(\vect u , \vect v) \mathrm{d}(\vect u , \vect v)$ and $\bar \Delta = \int_{[0,1]^{2d}} \Delta (\vect u , \vect v) \mathrm{d}(\vect u ,\vect v)$. Obviously, the function $\ell_n \mapsto \bar \Gamma / \ell_n^{4} + \bar \Delta \ell_n /n$ is minimized for 
\begin{align*}
\ell_{n,opt} = \Big(\frac{4 \bar \Gamma^2 }{\bar \Delta}\Big)^{1/5}n^{1/5},
\end{align*}
which can be considered as a theoretically optimal choice for the bandwidth parameter $\ell_n$.

In practice, the unknown quantities in $\ell_{n, opt}$ need to be estimated, namely $\sigma(\vect u, \vect v )= \sum_{k \in \mathbb{Z}} \gamma(k, \vect u, \vect v)$ and $M(\vect u , \vect v)=\sum_{k \in \mathbb{Z}} k^2 \gamma(k , \vect u, \vect v)$. For that purpose, we can closely follow \cite{BucKoj14} again. Let $L\in\{1, \dots, n\}$ be the smallest number such that the marginal autocorrelations at lag $L$ appear to be negligible, see \cite{BucKoj14} and \cite{PolWhi04} for details. Let $K$ denote the trapezoidal kernel, defined as $K(x )= [ \{ 2(1- \vert x \vert )\} \vee 0 ] \wedge 1 $. Then, set
\begin{align*}
\hat \sigma_n (\vect u, \vect v )& = \sum_{k=-L }^L K(k/L) \hat \gamma_n (k,\vect u ,\vect v), \quad
 \hat M_n (\vect u, \vect v) = \sum_{k = - L }^L K(k/L) k^2 \hat \gamma_n(k, \vect u , \vect v),
\end{align*}
where, for $\vect u , \vect v \in [0,1]^d$ and $k \in \{-L, \dots ,L\}$,
\begin{align*}
\hat \gamma_n (k, \vect u, \vect v ) = \begin{cases} 
n^{-1} \sum_{i=1}^{n-k} \hat f(\hat{ \vect U}_i, \vect u ) \hat f (\hat{ \vect U}_{i+k},\vect v),  \; & k \geq 0\\
								n^{-1} \sum_{i=1-k}^{n} \hat f(\hat{ \vect U}_i, \vect u ) \hat f (\hat {\vect U}_{i+k},\vect v),  \; &k < 0 \end{cases},
\end{align*}
and where
\begin{multline*}
\hat f(\hat {\vect U}_i , \vect u) 
= 
\ind (\hat {\vect U}_i \leq \vect u) - C_{n} (\vect u) - \sum_{\dimi=1}^d  C_{n}^{[\dimi]}(\vect u) \{ \ind (\hat U_{i \dimi} \leq u_\dimi) -u_\dimi \} \\
-\nabla C_{\theta_n}(\vect u) \{ J_{\theta_0}(\hat {\vect U}_i ) + \hat K_{ i,n,\theta_n}\}.
\end{multline*}
The plug-in principle finally yields an estimator $\hat \ell_{n,opt}$ for $\ell_{n,opt}$.
\newpage


\section{Runtime comparison} \label{runtimecomparison}
To compare the performance of our test to the test by \cite{RemPapSou12} in terms of runtime, we simulated $N=100$ test runs of  both tests under the null hypothesis for the Gaussian, the $t_4$ and the Clayton copula with $\tau=0.2$. The results can be found in Figure \ref{runtimes}, where we depict the relative runtimes $r_{pb}/r_{mb}$ of the test by \cite{RemPapSou12} ($r_{pb}$ for parametric bootstrap) compared to the runtime of the new test ($r_{mb}$ for multiplier bootstrap) as a function of the sample size (with $B=200$ bootstrap replications) and of the number of bootstrap replications (with sample size $n=200$), respectively.  In all considered scenarios the new test clearly outperforms the test by \cite{RemPapSou12} in terms of computational costs, up to a factor of almost $70$ for the Clayton copula with $B=500$ and $n=200$.

Note that the function corresponding to the Clayton copula in the left panel of Figure~\ref{runtimes} is the only decreasing function.
This is due to the fact that, for the Clayton copula and with increasing sample size, the evaluation of the score function in the data points in the multiplier bootstrap requires comparatively more time than the simulation of a Markovian copula process in the parametric bootstrap (even though all involved formulas are explicit).

\begin{figure}[h]
\includegraphics[width=.5\textwidth]{laufzeit_B=200}
\hspace{-2mm}
\includegraphics[width=.5\textwidth]{laufzeit_n=200}

\caption{Relative runtimes of the test by \cite{RemPapSou12} compared to the runtimes of the new testing procedure as a function of the sample size (left) and the number of bootstrap replications (right) for the Gaussian, the $t_4$ and the Clayton copula.}
\label{runtimes}
\end{figure}



\section{Remaining proofs }\label{sec:auxproofs}
\def\theequation{F.\arabic{equation}}
\setcounter{equation}{0}

\noindent


\begin{proof}[Proof of Lemma~\ref{lem:boot:hat}] As in the proof of Lemma~\ref{ml:le:01}, we may assume without loss of generality that $\Omega=\Omega_{n, \shaoc}$ for some $\shaoc>0$.
Let us write
$
L_{n2}^{\tbm} = A_n+B_n,
$
where
\begin{align*}
  A_n =\frac{1}{\sqrt{n}} \sum_{i=1}^n  Z_{i,n}^{(b)}\{J_{\theta_0}(\hat{\vect U}_i)-J_{\theta_0}({\vect U}_i )\}, \quad
  B_{n} =\frac{1}{\sqrt{n}} \sum_{i=1}^n  Z_{i,n}^{(b)}\{\hat{K}_{i,n,\theta_0}-{K}_{i,n,\theta_0}\},
\end{align*}
and consider each term separately. For $\eta\in(0,1/2)$, set $M_\eta=[\eta,1-\eta]^d$. By the mean value theorem, there exist intermediate values $\tilde{\vect U}_i$ between $\hat{\vect U}_i$ and $\vect U_i$ such that we can write $A_n=\sum_{\dimi=1}^d \{ A_{n,\dimi}(M_\eta) + A_{n,\dimi}(M_\eta^C) \}$, where, for any $M\subset[0,1]^d$,
\begin{align*}
  A_{n, \dimi}(M) &=
  \frac{1}{\sqrt{n}} \sum_{i=1}^n  Z_{i,n}^{(b)} J_{\theta_0}^{[\dimi]} (\tilde{\vect U}_i)
  (\hat{U}_{i,\dimi}-U_{i,\dimi}) \ind(\vect U_i \in M).
\end{align*} 
We begin with the treatment of $A_{n,\dimi}(M_\eta^C)$, for a fixed $\dimi$. Since we may assume $\Omega=\Omega_{n, \shaoc}$, we can bound $\| A_{n,\dimi}(M_\eta^C) \|$ by 
\begin{align*}
  \frac{\shaoc}{n} \sum_{i=1}^n \vert Z_{i,n}^{(b)}  \vert  q_\dimi(U_{i,\dimi}) 
  \sup_{\frac{\vert u- U_{i,\dimi}\vert }{q_\dimi(U_{i,\dimi})} \leq \frac{B}{\sqrt{n}}} \tilde{r}_\dimi(u) 
  \prod_{\dimi' \neq \dimi} \sup_{\frac{\vert u- U_{i,\dimi'}\vert }{q_{\dimi'}(U_{i,\dimi'})} \leq \frac{B}{\sqrt{n}}} 
  r_{\dimi'}(u) \ind(U_i \in M_\eta^C) 
\end{align*}
As in the proof of Lemma~\ref{ml:le:01}, by Condition~\ref{klasse2}, for sufficiently large $n$, the expectation of the latter expression converges to zero as $\eta\to 0$. Therefore, it remains to consider $A_{n,\dimi}(M_\eta)$ for fixed $\eta\in(0,1/2)$.

Since 
$\max_{i=1}^n \| \hat {\vect U}_{i} - \vect U_i \| \le \shaoc/\sqrt n$, we get that $A_{n,\dimi}(M_\eta) = \bar A_{n,\dimi}(M_\eta) +o_P(1)$ for $n\to\infty$, where
\[
  \bar A_{n,\dimi}(M_\eta) = \frac{1}{\sqrt{n}} \sum_{i=1}^n  Z_{i,n}^{(b)} J_{\theta_0}^{[\dimi]} (\tilde{\vect U}_i)
  (\hat{U}_{i,\dimi}-U_{i,\dimi}) \ind(\vect U_i \in M_\eta, \|\hat {\vect U}_i - \vect U_i \| \le \eta/2).
\]

By the Cauchy-Schwarz inequality, we obtain
\begin{multline*}
  \Exp\big[ \Vert  \bar A_{n,\dimi}(M_\eta) \Vert^2\big] 
  =
  \Eb\big[\{\bar A_{n,\dimi}(M_\eta)\}' \{\bar A_{n,\dimi}(M_\eta) \}\big] \\
  \le \sup_{\vect u \in M_{\eta/2}} \Vert J_{\theta_0}^{[\dimi]}(\vect u) \Vert^2 
  \times
  \Exp\big[\max_{i=1}^n | \hat U_{i, \dimi} - U_{i, \dimi} |^2 \big]
  \times 
  \frac{1}{n} \sum_{i,j=1}^n  \vert \mathbb{E}[ Z_{i,n}^{(b)}  Z_{j,n}^{(b)} ] |.
\end{multline*}
The first factor on the right-hand side is bounded by Condition~\ref{klasse2}.  The second factor is bounded by $\shaoc^2/n$ as we may assume that $\Omega=\Omega_{n, \shaoc}$. Regarding the third factor, note that  $\frac{1}{n} \sum_{i,j=1}^n  \vert \mathbb{E}[ Z_{i,n}^{(b)}  Z_{j,n}^{(b)} ] |=O(\ell_n)$ as shown in \eqref{eq:zzz}.  We can conclude that 
$\bar A_{n,\dimi}(M_\eta)$  is of order $O_p(n^{-1/4-\kappa/2}) =o_P(1)$, and therefore also $A_n=o_P(1)$.

For the proof of the lemma, it remains to be shown that $B_n=o_P(1)$. We can decompose
$
  B_n = \sum_{\dimi=1}^d (B_{n,1, \dimi} + B_{n,2, \dimi} + B_{n,3,\dimi}),
$
with 
\begin{align*}
  B_{n,1, \dimi} &= 
  \frac{1}{\sqrt{n}} \sum_{i=1}^n Z_{i,n}^{(b)} \frac{1}{n} \sum_{j=1}^n 
  \{ J_{\theta_0}^{[\dimi]}( \hat{\vect U}_j) -J_{\theta_0}^{[\dimi]}(\vect U_j)\} \{ \ind(U_{i,\dimi} \leq U_{j,\dimi})- {U}_{j,\dimi}\} \\
  B_{n,2, \dimi} &=
  \frac{1}{\sqrt{n}}\sum_{i=1}^n Z_{i,n}^{(b)} \frac{1}{n} \sum_{j=1}^n 
  \{ J_{\theta_0}^{[\dimi]}( \hat{\vect U}_j) -J_{\theta_0}^{[\dimi]}(\vect U_j)\} (U_{j,\dimi}- \hat{U}_{j,\dimi} ) \\
  B_{n,3, \dimi} &= \bigg\{ \frac{1}{\sqrt{n}} \sum_{j=1}^n J_{\theta_0}^{[\dimi]}(\vect U_j) (\hat{U}_{j,\dimi}-U_{j,\dimi}) \bigg\}
  \times \bigg\{ \frac{1}{n} \sum_{i=1}^n Z_{i,n}^{(b)}\bigg\}.
 \end{align*}
$B_{n,3,\dimi}$ converges in probability to zero: the first factor is of order $O_P(1)$ by a similar argumentation as before based on the fact that we may assume $\Omega=\Omega_{n, \shaoc}$, and the second factor is of order $O_P(\ell_n^{1/2}/n^{1/2})$ by \eqref{eq:zzz}.

Regarding $B_{n,1,\dimi}$, we can bound
\begin{multline*}
  \| B_{n,1,\dimi} \| \le 
  \bigg \{ \frac{1}{\sqrt{n}}\sup_{u \in [0,1]}\Big \vert 
  \sum_{i=1}^n Z_{i,n}^{(b)} \frac{ \ind (U_{i, \dimi}\leq u )-u}{q_\dimi(u)}\Big \vert \bigg \} \\
  \times 
  \bigg \{\frac{1}{n}\sum_{j=1}^n \Vert J_{\theta_0}^{[\dimi]}(\hat{\vect U}_j)- J_{\theta_0}^{[\dimi]}(\vect U_j) \Vert 
  q_\dimi(U_{j,\dimi}) \bigg\}.
\end{multline*}
The first factor on the right-hand side is of order $O_P(1)$ by Lemma \ref{weightedbootstrap}.
The second factor converges to $0$ in probability by the same argumentation as for the treatment of \eqref{eq:sni} in the proof Lemma~\ref{ml:le:01}.

A similar argumentation also works for $B_{n,2,\dimi}$: on the set $\Omega_{n, \shaoc}$, we have
\begin{align*}
  \Vert B_{n,2,\dimi} \Vert 
  \leq 
  \Big \{  \frac{1}{n} \sum_{i=1}^n \vert Z_{i,n}^{(b)} \vert \Big \}
  \times \Big \{\frac{\shaoc}{n} \sum_{j=1}^n  \Vert J_{\theta_0}^{[\dimi]}(\hat{\vect U}_j) -J_{\theta_0}^{[\dimi]}(\vect U_j) \Vert 
  q_\dimi(U_{j,\dimi}) \Big \},
\end{align*}
and this expression is $o_P(1)$ since $\mathbb{E}[\vert Z_{i,n}^{(b)}\vert ]<\infty$ and by the same reasons as in the proof of Lemma \ref{ml:le:01}. 
\end{proof}




\begin{proof}[Proof of Lemma~\ref{le:08}] Using Condition~\ref{cond:mixBK}, tightness of the vector of processes follows from marginal tightness of $\tilde \Bb_n$ and $\tilde \Bb_n^{(b)}$, see Theorem 3.1 in \cite{BucKoj14}. 


Regarding weak convergence of the finite dimensional distributions, we only consider $(\tilde{\mathbb{B}}_n,\tilde{\Theta}_n,\tilde{\mathbb{B}}_n^{(b)},\tilde{\Theta}_n^{(b)})$ for the ease of reading. 
By the Cram\'er-Wold device, we have to show that, for any $q,q'\in\N$, any $c_1,\dots ,c_q,\bar c_1,\dots ,\bar c_{q'} \in \mathbb{R}$, $c,\bar c \in \mathbb{R}^p$ and any $\vect u_1,\dots ,\vect u_q,\vect v_1,\dots ,\vect v_{q'} \in [0,1]^d$,
\begin{multline*}
V_n:=\sum_{s=1}^q c_s \tilde{\mathbb{B}}_n(\vect u_s) + \sum_{s=1}^{q'} \bar c_s \tilde{\mathbb{B}}_n^{(b)} (\vect v_s) + c' \tilde \Theta_n + \bar c' \tilde \Theta_n^{(b)} \\\weak
V:= \sum_{s=1}^q c_s \mathbb{B}_{C} (\vect u_s) + \sum_{s=1}^{q'} \bar c_s \mathbb{B}_{C}^{(b)}(\vect v_s) +c' \Theta+\bar c '\Theta^{(b)}.
\end{multline*}

First of all, we decompose \[V_n= \frac{1}{\sqrt{n}}\sum_{i=1}^n W_i+W_i^{(b)}+T_i+T_i^{(b)}\] with
$W_i=\sum_{s=1}^q c_s \{ \ind (\vect U_i \leq \vect u_s) - C_{\theta_0} (\vect u_s)\} ,
W_i^{(b)}=\sum_{s=1}^{q'} Z_{i,n}^{(b)} \bar c_s \{\ind( \vect U_i \leq \vect v_s)- C_{\theta_0}( \vect v_s)\} , 
T_i=c'\{ J_{\theta_0}(\vect U_i)+K_{i,\theta_0}\}$ and $
T_i^{(b)}=Z_{i,n}^{(b)}\bar c'\{ J_{\theta_0}(\vect U_i)+K_{i,\theta_0}\}. $  The subsequent proof is based on the `big block-small block'-technique. The assumption on $a$ in Condition~\ref{cond:mixBet} is equivalent to $\frac{1}{2(1+\nu)}< \frac{1}{2} - \frac{2+\nu}{a\nu}$ whence, noting that also $\kappa>\frac{1}{2(1+\nu)}$ by assumption, we may choose $0<\eta_1< \eta_2<\kappa$ such that $\frac{1}{2(1+\nu)}< \eta_1 < \eta_2 < \frac{1}{2} - \frac{2+\nu}{a\nu}$. Now, set $b_n=\lfloor n^{1/2-\eta_1} \rfloor$ (the length of the big blocks), $s_n=\lfloor n^{1/2-\eta_2} \rfloor$ (the length of the small blocks) and $k_n= \lfloor n/(b_n+s_n) \rfloor$ (the number of big or small blocks). Notice, that $k_n=O(n^{1/2+\eta_1})$. For $j=1, \dots , k_n$, set
\begin{align*}
B_{jn}&=\sum_{i=(j-1)(b_n+s_n)+1}^{(j-1)(b_n+s_n)+b_n} W_i+W_i^{(b)}+T_i+T_i^{(b)},\\
S_{jn}&=\sum_{i=(j-1)(b_n+s_n)+b_n+1}^{j(b_n+s_n)}W_i+W_i^{(b)}+T_i+T_i^{(b)},
\end{align*} 
such that we can write
\begin{align*}
V_n = \frac{1}{\sqrt{n}} \sum_{j=1}^{k_n} B_{jn}+ \frac{1}{\sqrt{n}} \sum_{j=1}^{k_n} S_{jn} +\frac{1}{\sqrt{n}} R_n,
\end{align*}
where $R_n=\sum_{i=k_n(b_n+s_n)+1}^{n} W_i+W_i^{(b)}+T_i+T_i^{(b)}$ is the sum over the remaining indices that are not part of a big or a small block. 

First of all, let us show that the variance of $V_n$ is equal to $\Var(\frac{1}{\sqrt{n}} \sum_{j=1}^{k_n}B_{jn}) +o(1)$ as $n\to\infty$, i.e., that 
\begin{multline} 
\frac{1}{n}\Var\Big(\sum_{j=1}^{k_n} S_{jn}\Big) 
+ \frac{1}{n}\Var(R_n) 
+ \frac{2}{n} \sum_{j,j'=1}^{k_n} \Cov(B_{jn},S_{j'n}) \\
+ \frac{2}{n} \sum_{j=1}^{k_n} \Cov(B_{jn},R_n) 
+ \frac{2}{n} \sum_{j=1}^{k_n} \Cov(S_{jn},R_n) \label{var:decomp}
\end{multline}
vanishes as $n \to \infty$. To this end, we will frequently exploit the following bounds which are consequences of Lemma 3.9 and Lemma 3.11 in \cite{DehPhi02}
\begin{align*} 
\vert \mathbb{E}[W_iW_{i'}] \vert 
&=
\Big \vert  \sum_{s,s'=1}^q c_s c_{s'}  \mathbb{E}[\{\ind(\vect U_i \leq \vect u_s)-C_{\theta_0}(\vect u_s)\}\{\ind(\vect U_{i}\leq \vect u_{s'})-C_{\theta_0 }( \vect u _{s'})\}] \Big \vert  \\
&\leq  
4\sum_{s,s'=1}^q \vert c_s c_{s'} \vert \alpha (\vert i - i' \vert)  \le \const \times \alpha ( \vert i - i' \vert )\\
\vert \mathbb{E}[W_iT_{i'}] \vert 
&=
\Big \vert \sum_{s=1}^q c_s c'  \mathbb{E}[\{ \ind (\vect U_i \leq \vect u_s)-C_{\theta}(\vect u_s)\}J_{\theta_0}(\vect U_{i'})] \Big \vert\\
&\leq 
20\Vert c \Vert  \sum_{s=1}^q \vert c_s \vert  \alpha (\vert i - i' \vert)^{\frac{1+\nu}{2+\nu}} \Exp [ \Vert J_{\theta_0} \Vert^{2+\nu}]^{\frac{1}{2+\nu}} \le \const \times \alpha ( \vert i - i' \vert )^{\frac{1+\nu}{2+\nu}}\\
\vert \mathbb{E}[T_i T_{i'}] \vert 
&= 
\vert c' \mathbb{E}[ J_{\theta_0}(\vect U_i)J_{\theta_0}(\vect U_{i'})']c \vert\leq 40 \Vert c \Vert^2 \alpha(\vert i - i'\vert )^{\frac{\nu }{2+\nu}} \Exp [ \Vert J_{\theta_0} \Vert^{2+\nu}]^{\frac{2}{2+\nu}} \\
&\le \const \times 
\alpha(\vert i - i'\vert )^{\frac{\nu}{2+\nu}}.
\end{align*} 
Analogously, $\vert \mathbb{E}[W_i^{(b)}W_{i'}^{(b)}]
\vert \le \const \times \alpha ( \vert i - i' \vert ))$, $\vert \mathbb{E}[W_i^{(b)}T_{i'}^{(b)}] \vert \le \const \times \alpha(\vert i-i'\vert)^{\frac{1+\nu}{2+\nu}}$ and
 $\vert \mathbb{E}[T_i^{(b)} T_{i'}^{(b)}] \vert \le \const \times \alpha(\vert i - i'\vert )^{\frac{\nu}{2+\nu}}$. Notice that all the other pairs of random variables are uncorrelated and that the largest bound is a constant multiple of $\alpha(\vert i - i'\vert )^{\frac{\nu}{2+\nu}}$.
Now, we can begin with the discussion of the first summand in (\ref{var:decomp}). We have 
\begin{align}
\label{eq:S}
\frac{1}{n}\Var \Big(\sum_{j=1}^{k_n} S_{jn}\Big) 
=
\frac{1}{n} \sum_{j=1}^{k_n} \Var(S_{jn})+ \frac{2}{n} \sum_{j \neq j'} \Cov (S_{jn},S_{j'n}).
\end{align}
Since the distance between any two summands in $S_{jn}$ and $S_{j'n}$ for $j\ne j'$ is at least $b_n$, their covariance is of order $\alpha(b_n)^{\frac{\nu}{2+\nu}}=O(b_n^{-a{\nu}/({2+\nu})})$. 
Observing that $S_{jn}$ consists  of  $s_n$  summands, we obtain that the second term in the last display is of order $O(k_n^2 s_n^2 b_n^{-a{\nu}/({2+\nu})}n^{-1})=O(n^{1-a {\nu}/({4+2\nu})+\eta_1(2+a {\nu}/({2+\nu}))-2\eta_2})=O(n^{1-a {\nu}/({4+2\nu})+\eta_2a {\nu}/({2+\nu})})=
o(1)$
 since, by construction,  $\eta_1 < \eta_2< \frac{1}{2} - \frac{2+\nu}{a\nu}$.
For the first sum on the right-hand side of \eqref{eq:S}, we have, by dominated convergence, 
\begin{align*}
\frac{1}{n}\sum_{j=1}^{k_n}\Var(S_{nj})=\frac{1}{n}\sum_{j=1}^{k_n}\mathbb{E}[S_{nj}^2] 
\leq&\ \const \times \frac{1}{n} \sum_{j=1}^{k_n} \sum_{i=-s_n}^{s_n} (s_n-\vert i \vert)\alpha(\vert i \vert)^{\frac{\nu }{2+\nu}}\\
= & \ O(k_ns_nn^{-1})= O(n^{\eta_1-\eta_2})=o(1).
\end{align*}

For the second term of \eqref{var:decomp}, we have 
\begin{align*}
\frac{1}{n} \Var (R_n) 
%=&\ \frac{1}{n} \Var \Big(\sum_{i=k_n(b_n+s_n)+1}^n W_i+W_i^{(b)}+T_i+T_i^{(b)} \Big) \\
\leq &\ \const \times \frac{1}{n} \sum_{i,i'=k_n(b_n+s_n)+1}^n \alpha(\vert i-i' \vert)^{\frac{\nu}{2+\nu}}\\
\leq &\ \const \times \frac{1}{n}\sum_{i=- \{n-k_n(b_n+s_n) \}}^{n-k_n(b_n+s_n)}(n-k_n(s_n+b_n )-\vert i \vert )\alpha(\vert i \vert)^{\frac{\nu}{2+\nu}} \\
 =& \ O( \{n-k_n(b_n+s_n)\}/n) = O((b_n+s_n)/{n})=o(1),
\end{align*}
where we used that 
%$n-\{k_n(b_n+s_n) \} \leq n - (\frac{n-b_n-s_n+1}{b_n+s_n}(b_n+s_n)) =b_n+s_n+1$. 
$k_n \ge n/(b_n+s_n) - 1$.

Now, let us bound the third term in \eqref{var:decomp}.  First we notice that, if $j=j'$ or $j'=j-1$, we have $\vert \mathbb{E}[B_{jn}S_{j'n}]\vert \leq \const\sum_{i=1}^{b_n} \sum_{i'= b_n+1 }^{s_n+b_n}\alpha(\vert i-i' \vert)^{\frac{\nu}{2+\nu}} \leq\const \sum_{i=1}^{s_n+b_n} i \alpha (i)^{\frac{\nu}{2+\nu}}\le \const< \infty$, since $a\nu/(2+\nu)>2+1/\nu$ by Condition~\ref{cond:mixBet}. In the other cases the distance between the blocks $B_{jn}$ and $S_{j'n}$ is at least $b_n$, such that $\vert \mathbb{E}[B_{jn}S_{j'n} ] \vert =O(b_ns_n\alpha({b_n})^{\frac{\nu}{2+\nu}})$.  Together, this yields 
\begin{multline*}
\frac{2}{n} \sum_{j,j'=1}^{k_n} \mathbb{E}[B_{jn}S_{j'n} ] =O(n^{-1}k_n) + O(k_n^2b_n s_n b_n^{-a\frac{\nu}{2+\nu}}n^{-1})\\
=O(n^{-1/2+\eta_1}) + O(n^{1-\frac{a}{2}\frac{ \nu}{2+\nu}+\eta_1(1+a\frac{\nu}{2+\nu})-\eta_2})
=o(1),
\end{multline*}
where the last equality follows exactly as above for the treatment of the first summand in \eqref{var:decomp}.
In the same manner, we get   $\frac{2}{n} \sum_{j=1}^{k_n } \mathbb{E} [ B_{jn}R_n] =  O(n^{\frac{1}{2}-\frac{a}{2}\frac{\nu }{2+\nu}-\eta_1 +a \frac{\nu }{2+\nu}\eta_2})=o(1)$  
and $\frac{2}{n} \sum_{j=1}^{k_n} \mathbb{E}[S_{nj}R_n] =O(n^{\frac{1}{2}-\frac{a}{2}\frac{\nu }{2+\nu}-\eta_2+a\frac{\nu }{2+\nu}\eta_1})+O(\{b_n+s_n\}/n)=o(1)$. 




For the next step of the proof, let $B'_{jn}, j=1, \dots, k_n$ denote independent random variables such that each $B'_{jn}$ has the same distribution as $B_{jn}$. We will show that the characteristic function of $n^{-1/2} \sum_{j=1}^{k_n} B_{jn}$ is asymptotically equivalent to the characteristic function of $n^{-1/2} \sum_{j=1}^{k_n} B'_{jn}$. For $t\in\R$, define $\Psi_{jn}(t)= \exp( \text{i}t n^{-1/2}B_{jn})$ and notice that $\mathbb{E}[\prod_{j=1}^{k_n} \Psi_{jn}(t)]$ and $\prod_{j=1}^{k_n} \mathbb{E}[\Psi_{jn}(t)]$ are the characteristic functions of $n^{-1/2} \sum_{j=1}^{k_n} B_{jn}$  and  $n^{-1/2}\sum_{j=1}^{k_n} B'_{jn}$, respectively.
The difference of the two characteristic functions can be decomposed as follows
\begin{multline*}
\Big \vert \mathbb{E} \Big[\prod_{j=1}^{k_n} \Psi_{jn}(t)\Big ]- \prod_{j=1}^{k_n} \mathbb{E}[\Psi_{jn }(t)] \Big \vert  \leq \Big \vert \mathbb{E}\Big[\prod_{j=1}^{k_n} \Psi_{jn}(t)\Big] - \mathbb{E}[\Psi_{1n}(t)] \mathbb{E}[\prod_{j=2}^{k_n} \Psi_{jn}(t)] \Big \vert \\
\hspace{30mm}+ \vert \mathbb{E}[\Psi_{1n}(t)] \vert \times \Big\vert \mathbb{E}\Big[\prod_{j=2}^{k_n} \Psi_{jn}(t)\Big ] -\mathbb{E}[\Psi_{2n}(t) ] \mathbb{E}[\prod_{j=3}^{k_n}\Psi_{jn}(t)] \Big\vert  \\
+ \dots  + \Big \vert \prod_{j=1}^{k_n-2 } \mathbb{E}[\Psi_{jn}(t)] \Big \vert \times \Big \vert \mathbb{E}\Big [\prod_{k_n-1}^{k_n} \Psi_{jn}(t) \Big]-\prod_{j=k_n-1}^{k_n} \mathbb{E} [\Psi_{jn}(t)]\Big\vert 
\end{multline*} 
Applying Lemma 3.9 in \cite{DehPhi02} $(k_n-1)$ times, we get
\begin{multline*}
\Big\vert \mathbb{E}\Big[\prod_{j=1}^{k_n} \Psi_{jn}(t)\Big] - \prod_{j=1}^{k_n} \mathbb{E}[\Psi_{jn}(t)] \Big \vert \\
\leq 2 \pi\times (k_n -1 ) \max_{i=1,\dots ,k_n-1} \alpha(\sigma(\Psi_{in}),\sigma\{\prod_{j=i+1}^{k_n}\Psi_{jn}(t)\}) ,
\end{multline*}
which is of order $O(k_n\alpha(s_n))=O(n^{1/2-a/2+a\eta_2+\eta_1})$. Using that $\eta_1<\eta_2$, this expression converges to $0$ by the choice of $\eta_2$  and the fact that $(2+v)/(a\nu) > 1/a >1/(a+1)$.
As a consequence, provided $n^{-1/2} \sum_{j=1}^{k_n} B'_{jn}$ converges weakly, then so does $n^{-1/2} \sum_{j=1}^{k_n} B_{jn}$ with the same limiting distribution. 

Therefore,  in order to finalize the proof, it remains to be shown that $n^{-1/2} \sum_{j=1}^{k_n} B'_{jn}$ converges weakly to $V$. This will be accomplished by 
proving that the variance of $n^{-1/2}\sum_{j=1}^{k_n} B'_{jn}$ converges to $\Var(V) $ as $n \to \infty$ and that the Lindeberg-condition from the Lindeberg-Feller central limit theorem for independent triangular arrays is met. We begin with the convergence of the variance and note that
\begin{align*}%\label{variance}
\Var(V)=& \sum_{s,s'=1}^q c_s c_{s'} \sum_{i \in \mathbb{Z}} \gamma (i,\vect u_s , \vect u_{s'}) + \sum_{s,s'=1}^{q'} \bar c_s \bar c_{s'} \sum_{i \in \mathbb{Z}} \gamma(i,\vect v_s , \vect v_{s'}) \\
&+ \sum_{s=1}^q c_s c'\sum_{\i \in \mathbb{Z}} \bar \gamma (i,\vect u_s ) + \sum_{s=1}^{q'} \bar c_s \bar c'\sum_{i \in \mathbb{Z}} \bar \gamma (i,\vect v_s )  \\
& +  c' \sum_{i \in \mathbb{Z}} \tilde \gamma (i)c +  \bar c'\sum_{i \in \mathbb{Z}} \tilde \gamma (i)\bar c   ,
\end{align*}
where $\gamma(i,\vect u,\vect v)= \Cov \{ \ind(\vect U_1 \leq \vect u), \ind(\vect U_{1+i} \leq \vect v)\}$, $ \bar \gamma (i,\vect u ) = \Cov \{ J_{\theta_0}(\vect U_1) + K_{1, \theta_0} , \ind( \vect U_{1+i} \leq \vect u) \}  $ and $\tilde \gamma (i) = \Cov \{ J_{\theta_0}(\vect U_1) + K_{1, \theta_0}, J_{\theta_0}(\vect U_{1+i} ) + K_{1+i,\theta_0} \}$.
Now, let us show that $\Var ( \frac{1}{\sqrt{n}} \sum_{j=1}^{k_n}B'_{jn})=\Var (V_n) + o(1)$. To this end, note that 
\begin{multline*}
\Var\Big(n^{-1/2}\sum_{j=1}^{k_n}B'_{jn}\Big)= \frac{1}{n} \sum_{j=1}^{k_n} \Var(B'_{jn})= \frac{1}{n} \sum_{j=1}^{k_n} \Var(B_{jn})\\
 =\Var  \Big(n^{-1/2}\sum_{j=1}^{k_n} B_{jn}\Big) - \frac{1}{n}\sum_{j\neq j'} \Cov(B_{jn},B_{j'n}).
\end{multline*}
Since we have already shown in the beginning of the proof that the variance on the right-hand side equals $\Var(V_n)+o(1)$, it remains to be shown that $\frac{1}{n}\sum_{j\neq j'} \Cov(B_{jn},B_{j'n})=o(1)$. Since the distance between the random variables within the two blocks is at least $s_n$, we have $\mathbb{E}[B_{jn}B_{j'n}]=O(b_n^2 \alpha({s_n})^{\frac{\nu }{2+\nu}})$ for $j \neq j'$. Therefore, $n^{-1}\sum_{j\neq j'} \Cov(B_{jn},B_{j'n}) = O(n^{-1}k_n^2b_n^2s_n^{-a\frac{\nu}{2+ \nu}})=O(n^{1-\frac{a}{2} \frac{\nu}{2+\nu}+a \frac{\nu}{2+ \nu}\eta_2})$, 
which is 
$o(1)$ as shown above.

Now, let us show that $\Var(V_n) \to \Var(V)$ as $n \to \infty$.  We can write $\Var(V_n)$ as 
\begin{align}
\label{eq:varVn}
& \frac{1}{n}\sum_{i,i'=1}^{n} \bigg[ 
\bigg\{ \sum_{s,s'=1}^q  c_sc_{s'}  \gamma(i-i',\vect u_s, \vect u_{s'}) 
+ \sum_{s,s'=1}^{q'}\bar c_s \bar c_{s'} \dmd \{(i-i')/\ell_n\}\gamma(i-i',\vect v_s,\vect v_{s'}) \bigg\} \nonumber \\
&\hspace{2cm}
+\bigg\{\sum_{s=1}^q  c' c_s \bar \gamma (i - i', \vect u_s) 
+\sum_{s=1}^{q'}\bar c' \bar c_s  \dmd \{(i-i')/\ell_n\}\bar \gamma (i - i', \vect v_s) \bigg\}  \nonumber \\
&\hspace{4cm}
+ \Big\{ c' \tilde \gamma (i - i') c +  \dmd \{(i-i')/\ell_n\} \bar c'\tilde \gamma (i - i') \bar c \Big\} \bigg]  \nonumber \\
=&  \sum_{i=-n}^n \frac{n-\vert i \vert }{n} \bigg[ \bigg\{
\sum_{s,s' =1}^q c_s c_{s'}\gamma(i, \vect u_s , \vect u _{s'}) + \sum_{s,s'=1}^{q'} c_s c_{s'} \dmd (i/\ell_n)  \gamma(i, \vect v_s, \vect v_{s'}) \bigg\}  \nonumber \\
& \hspace{2cm}
+\bigg\{ \sum_{s=1}^q c'  c_s  \bar \gamma (i , \vect u_s)  +\sum_{s=1}^{q'}\bar c' \bar c_s  \dmd (i /\ell_n) \bar \gamma (i , \vect v_s)  \bigg\}  \nonumber \\
&\hspace{4cm}
+\Big\{ c'\tilde \gamma (i)c + \dmd (i/\ell_n)\bar c'\tilde \gamma (i) \bar c\Big\} \bigg].
\end{align}
For the sake of brevity, we will only show convergence of the terms in the first curly brackets 
on the right-hand side of the last display 
to the respective terms in $\Var(V)$.  
We have 
\begin{multline*}
\sum_{i=-n}^n \frac{n-\vert i \vert }{n}\sum_{s,s' =1}^q c_s c_{s'} \gamma(i, \vect u_s , \vect u _{s'})\\
= \sum_{s,s' =1}^q c_s c_{s'} \sum_{i=-n}^n \gamma(i, \vect u_s , \vect u _{s'})-\sum_{s,s' =1}^q c_s c_{s'} \frac{1}{n}\sum_{i=-n}^n \vert i \vert  \gamma(i, \vect u_s , \vect u _{s'}).
 \end{multline*}
Since $\vert \gamma(i ,\vect u, \vect v ) \vert \leq \const \times \alpha(|i|)$ and  $\sum_{i=1}^\infty \vert i \vert \alpha (i)<\infty $, the second term on the right-hand side vanishes as  $n \to \infty$, whereas the first term converges to $ \sum_{s,s' =1}^q c_s c_{s'} \sum_{i \in \mathbb{Z}} \gamma (i, \vect u_s , \vect u _{s'})$.

Moreover, as $\dmd(h)=0$ for $\vert h \vert >1$ and $\ell_n=o(n)$, we have
\begin{multline*} 
\sum_{i=-n}^n \frac{n- \vert i \vert }{n} \sum_{s,s'=1}^{q'} \bar c_s \bar c_{s'} \dmd (i/\ell_n)  \gamma(i, \vect v_s, \vect v_{s'}) \\
= \sum_{s,s'=1}^{q'} \bar c_s \bar c_{s'} \sum_{i \in \mathbb{Z}} \frac{n- \vert i \vert }{n} \dmd (i/\ell_n) \gamma(i, \vect v_s, \vect v_{s'}),
\end{multline*}
By continuity of $\dmd$ in $0$, we have $\frac{n-\vert i \vert}{n} \dmd (i /\ell_n) \to 1$  as $n \to \infty$ for any fixed $i \in \mathbb{Z}$.  Moreover, 
$\vert \frac{n-\vert i\vert}{n}  \dmd (i/\ell_n) \gamma(i, \vect v_s ,\vect v_{s'})\vert \leq \const \times \alpha(i)$ for all $i \in \mathbb{Z}$ and all $s,s'=1,\dots ,{q'}$. Therefore, by dominated convergence, as $n\to\infty$,
\begin{align*}
\sum_{s,s'=1}^{q'} \bar c_s \bar c_{s'} \sum_{i=-n}^n \frac{n- \vert i \vert }{n} \dmd (i/\ell_n)  \gamma(i, \vect v_s, \vect v_{s'})\to \sum_{s,s'=1}^{q'} \bar c_s \bar c_{s'} \sum_{i \in \mathbb{Z}} \gamma(i, \vect v_s, \vect v_{s'}).
\end{align*}


The convergence of the remaining summands in \eqref{eq:varVn} follows along similar lines, exploiting that $\Vert \bar \gamma (i ,\vect u ) \Vert \leq \const \times \alpha(i)^{\frac{1+\nu}{2+\nu}}$, $\Vert \tilde \gamma (i) \Vert_{op} \leq \const \times \alpha(i)^{\frac{\nu}{2+\nu}}$ and that $\sum_{i=1}^\infty \vert i \vert \alpha(i)^{\frac{\nu}{2+\nu} }<\infty$.


Finally, let us prove the Lindeberg condition, i.e., that, for any $ \eps >0$,
\begin{align*}
\frac{1}{n } \sum_{j=1}^{k_n} \Exp[B'^2_{jn}  \ind(\vert B'_{jn} \vert > \sqrt{n} \eps) ] = \frac{1}{n }\sum_{j=1}^{k_n} \Exp[B_{jn}^2 \ind(\vert B_{jn} \vert > \sqrt{n} \eps) ] \to 0
\end{align*}
as $n \to \infty$. To bound the former expression we use H\"older's inequality with 
%$\frac{1}{p}=\frac{2}{2+\nu}$ and $\frac{1}{q}=\frac{\nu}{2+\nu}$ 
$p=(2+\nu)/2$ and $q=(2+\nu)/\nu$
and Markov's inequality to obtain
\begin{align*}
\frac{1}{n } \sum_{j=1}^{k_n} \mathbb{E}[B_{jn}^2 \ind(\vert B_{jn } \vert \leq \sqrt{n} \eps )] 
& \leq 
\frac{1}{n} \sum_{j=1}^{k_n} \mathbb{E}[\vert B_{jn} \vert ^{2+\nu}]^{\frac{2}{2+\nu}} \mathbb{E}[ \ind (\vert B_{jn} \vert  \leq \sqrt{n} \eps)^{\frac{2+\nu}{\nu}}]^{\frac{\nu }{2+ \nu}}\\
&=  
\frac{1}{n} \sum_{j=1}^{k_n} \mathbb{E}[\vert B_{jn} \vert^{2+\nu}]^{\frac{2}{2+\nu}} \Prob(\vert B_{jn} \vert  \leq \sqrt{n} \eps)^{\frac{\nu }{2+ \nu}}\\
& \leq  
\frac{1}{n} \sum_{j=1}^{k_n} \mathbb{E}[\vert B_{jn} \vert^{2+\nu}]^{\frac{2}{2+\nu}}\mathbb{E} [\vert B_{jn} \vert^{2+\nu}]^{\frac{\nu}{2+\nu}}(\sqrt{n}\eps)^{-\nu}\\
&= 
\frac{1}{n} \sum_{j=1}^{k_n} \mathbb{E}[\vert B_{jn} \vert^{2+\nu}] (\sqrt{n}\eps)^{-\nu}
\end{align*} 
By Minkowski's inequality, we can bound $\mathbb{E}[\vert B_{jn} \vert^{2+\nu}]^{\frac{1}{2+\nu}}$ by a sum over $b_n$ summands of the form $\mathbb{E}[\vert W_i \vert^{2+\nu} ]^{\frac{1}{2+\nu}} + \mathbb{E}[\vert W_i^{(b)} \vert^{2+\nu}]^{\frac{1}{2+\nu}} +\mathbb{E}[\vert T_i \vert^{2+\nu}]^{\frac{1}{2+\nu}} +\mathbb{E}[\vert T_i^{(b)} \vert^{2+\nu}]^{\frac{1}{2+\nu}}
$, whence $\mathbb{E}[\vert B_{jn} \vert^{2+\nu}]=O(b_n^{2+\nu})$.
This finally implies
\begin{align*}
\frac{1}{n } \sum_{j=1}^{k_n} \mathbb{E}[B_{jn}^2 \ind(\vert B_{jn } \vert \leq \sqrt{n} \eps )]=O(b_n^{2+\nu}k_n n^{-\nu/2-1})=O(n^{1/2-\eta_1(1+\nu)})=o(1),
\end{align*}
by the definition of $\eta_1$ and the Lemma is proved.
\end{proof}


\begin{proof}[Proof of Lemma~\ref{lem:taylor}]
It follows from the mean value theorem that
\begin{align*}
\sqrt{n} \{ C_{\theta_n}(\vect u) -C_{\theta_0}(\vect u)\}
							&= \nabla {C}_{\theta_0}(\vect u) \Theta_n + R_n(\vect u),
\end{align*}
where  $R_n(\vect u )=\sqrt{n}\sum_{s=1}^p  \big\{ \frac{\partial}{\partial \theta_s } C_{\tilde{\theta}}(\vect u) - \frac{\partial}{\partial \theta_{s} } C_{{\theta}_0}(\vect u)  \big\}  (\theta_{ns} -\theta_{0s})$ and where $\tilde{\theta}$ denotes an intermediate point lying between $\theta_0$ and $\theta_n$.  The Cauchy-Schwarz inequality allows to estimate 
\begin{align*}
\sup_{\vect u  \in [0,1]^d}\Vert R_n(\vect u) \Vert 
\leq  \sup_{\Vert \theta' - \theta_0 \Vert < \Vert \theta_n - \theta_0 \Vert} \sup_{\vect u \in [0,1]^d } \Vert \nabla {C}_{\theta'}(\vect u)- \nabla {C}_{\theta_0}(\vect u) \Vert  \times \Vert \Theta_n \Vert .
\end{align*}
Since $\Theta_n =\tilde{ \Theta}_n+o_p(1)$ for $n\to\infty$ as a consequence of Lemma~\ref{ml:le:01} and~\ref{ml:le:04}, we obtain that $\Vert \Theta_n \Vert =O_P(1)$ from Lemma~\ref{le:08} (note that Condition~\ref{cond:mixBK} was used in Lemma~\ref{le:08} only for the tightness part, whence we do not need to assume it here). Fix $\eta >0$. By Condition \ref{cond:contdotC}, we may choose $\delta>0$ such that  $ \sup_{\Vert \theta' - \theta_0 \Vert < \delta} \sup_{\vect u \in [0,1]^d } \Vert \nabla{C}_{\theta'}(\vect u)- \nabla{C}_{\theta_0}(\vect u) \Vert \leq  \eta$. Therefore, 
\begin{align*} 
%\label{est:grad}
 \Prob(\sup_{\vect u \in [0,1]^d } \sup_{\Vert \theta' - \theta_0 \Vert < \Vert \theta_n - \theta \Vert} \Vert 
	\nabla{C}_{\theta'}(\vect u)- \nabla{C}_{\theta_0}(\vect u) \Vert > \eta)
\leq  \Prob(\Vert \theta_n - \theta_0 \Vert > \delta),
\end{align*}
which converges to $0$ as $n \to \infty$, since $\theta_n$ is a consistent estimator of $\theta_0$. 
\end{proof}

\noindent
\textbf{Acknowledgements.} 
The authors would like to thank three unknown referees and two editors for their constructive comments on an earlier version of this manuscript, which led to a substantial improvement of the paper.
%The author is also thankful to \textit{Ivan Kojadinovic} for thorough proofreading and numerous suggestions concerning this manuscript. 
%Research supported by the IAP research network grant P7/06 of the Belgian government (Belgian Science Policy) is gratefully acknowledged
The authors are also thankful to Brendan K.\ Beare for kindly providing us with the gasoline price markup data and Ivan Kojadinovic for helpful discussions regarding this manuscript.

Parts of this paper were written when Axel B\"ucher was a post-doctoral researcher at Universit\'e catholique de Louvain, Belgium, a visiting professor at Universit\"at Heidelberg, Germany, and a visiting professor at Technische Universit\"at Dortmund, Germany.

This work has been supported in parts by the Collaborative Research Center ``Statistical modeling of nonlinear dynamic processes'' (SFB 823) of the German Research Foundation (DFG) and by the IAP research network Grant P7/06 of the Belgian government (Belgian Science Policy), which is gratefully acknowledged.\\

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