\begin{algorithm}
  \caption{$L$-lag-test (empirical version)}
  \label{alg:llt}
  \begin{algorithmic}[1]
    \Require $(X_1, \ldots, X_n)$, $(Y_1, \ldots, Y_n)$, $d$, $\alpha$, $L$, $B$
    \For{$k = 1, \ldots, n-L$}
    \State $X'_k \gets \left(X_k \ldots, X_{k+L}\right)$
    \State $Y'_k \gets \left(Y_k, \ldots, Y_{k+L}\right)$
    \EndFor
    \State $N \gets \lfloor (n-L)/d\rfloor$
    \For{$b = 1, \ldots, B$}
    \State $\left(X^*, Y^*\right) \gets \texttt{IBB}\left((X'_1, \ldots, X'_{n-L}), (Y'_1, \ldots, Y'_{n-L}), d\right)$
    \State $D_b \gets Nd \, \mathrm{dcov}_{n-L}\left(X^*, Y^*\right)$\label{algline:dcov1}
    \EndFor
    \State $c_\alpha^* \gets$ empirical upper $\alpha$-quantile of $\{D_1, \ldots, D_B\}$
    \If{$(n-L) \, \mathrm{dcov}_{n-L}(X', Y') > c_\alpha^*$}\label{algline:dcov2}
    \State $\texttt{Decision} \gets$ `Reject $H_0$'
    \Else
    \State $\texttt{Decision} \gets$ `Do not reject $H_0$'
    \EndIf
    \Ensure \texttt{Decision}
  \end{algorithmic}
\end{algorithm}

\begin{algorithm}
\caption{$L$-lag-test (empirical version)}
\begin{algorithmic}
[1]
\Require $(X_1, \ldots, X_n)$, $(Y_1, \ldots, Y_n)$, $d$, $\alpha$, $L$, $B$
\For{$k = 1, \ldots, n-L$}
\State $X'_k \gets \left(X_k \ldots, X_{k+L}\right)$
\State $Y'_k \gets \left(Y_k, \ldots, Y_{k+L}\right)$
\EndFor
\State $N \gets \lfloor (n-L)/d\rfloor$
\For{$b = 1, \ldots, B$}
\State $\left(X^*, Y^*\right) \gets \texttt{IBB}\left((X'_1, \ldots, X'_{n-L}), (Y'_1, \ldots, Y'_{n-L}), d\right)$
\State $D_b \gets Nd \, \mathrm{dcov}_{n-L}\left(X^*, Y^*\right)$ \EndFor
\State $c_\alpha^* \gets$ empirical upper $\alpha$-quantile of $\{D_1, \ldots, D_B\}$
\If{$(n-L) \, \mathrm{dcov}_{n-L}(X', Y') > c_\alpha^*$} \State $\texttt{Decision} \gets$ `Reject $H_0$'
\Else
\State $\texttt{Decision} \gets$ `Do not reject $H_0$'
\EndIf
\Ensure \texttt{Decision}
\end{algorithmic}
\end{algorithm}
{
    "title": "a bootstrap test for independence of time series based on the distance   covariance",
    "id": "2112.14091",
    "abstract": "we present a test for independence of two strictly stationary time series based on a bootstrap procedure for the distance covariance. our test detects any kind of dependence between the two time series within an arbitrary maximum lag $l$. in simulation studies, our test outperforms alternative testing procedures. in proving the validity of the underlying bootstrap procedure, we generalise bounds for the wasserstein distance between an empirical measure and its marginal distribution under the assumption of $\\alpha$-mixing. previous results of this kind only existed for i.i.d. processes.",
    "categories": "math.st stat.th",
    "doi": "",
    "created": "2021-12-28",
    "updated": "2024-02-05",
    "authors": [
        "annika betken",
        "herold dehling",
        "marius kroll"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2112.14091"
}
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