Test for independence of long-range dependent time series using distance covariance

TL;DR

The paper addresses testing independence between two long-range dependent time series by extending distance covariance to a lag-cross-covariance framework. It develops a Hilbert-space formulation for empirical distance cross-covariances and derives a non-central limit theorem for Hilbert-space-valued LRD processes, enabling the asymptotic analysis of a linear statistic $T_n=\sum_{h\ge0} a_h \,{\operatorname{dcov}}_n(X,Y;h)$. Since the limiting distribution is unknown under long-range dependence, the authors propose a subsampling procedure and prove its validity, establishing consistency of the test against broad alternatives. The paper also provides extensive finite-sample evidence, showing improved power over covariance-based methods for nonlinear dependencies, and demonstrates practical applicability with cross-river hydrological data, highlighting the method’s relevance for detecting complex cross-dependencies in environmental time series. Overall, the work contributes both theoretical advancements in Hilbert-space limit theory for LRD processes and a practical testing tool for independence in dependent time-series settings.

Abstract

We apply the concept of distance covariance for testing independence of two long-range dependent time series. As test statistic we propose a linear combination of empirical distance cross-covariances. We derive the asymptotic distribution of the test statistic, and we show consistency against a very general class of alternatives. The asymptotic theory developed in this paper is based on a novel non-central limit theorem for stochastic processes with values in an $L^2$-Hilbert space. This limit theorem is of general theoretical interest which goes beyond the context of this article. Subject to the dependence in the data, the standardization and the limit distributions of the proposed test statistic vary. Since the limit distributions are unknown, we propose a subsampling procedure to determine the critical values for the proposed test, and we provide a proof for the validity of subsampling. In a simulation study, we investigate the finite-sample behavior of our test, and we compare its performance to tests based on the empirical cross-covariances. As an application of our results we analyze the cross-dependencies between mean monthly discharges of three rivers.

Figures (11)

• Figure 1: "Linearly" correlated data $(X_i,Y_i)$, $i=1, \ldots, 500$, with parameters $H$ and $r$.
• Figure 2: "Parabolically" correlated data $(X_i,Y_i)$, $i=1, \ldots, 500$, with parameters $H$ and $v$.
• Figure 3: "Wavily" correlated data $(X_i,Y_i)$, $i=1, \ldots, 500$, with parameters $H$ and $v$.
• Figure 4: "Rectangularly" correlated data $(X_i,Y_i)$, $i=1, \ldots, 500$, with parameters $H$ and $v$.
• Figure 5: Rejection rates of the hypothesis tests resulting from the empirical covariance and the empirical distance covariance obtained by subsampling based on "linearly" correlated fractional Gaussian noise time series $X_j$, $j=1, \ldots, n$, $Y_j$, $j=1, \ldots, n$ with block length $l_n~=~\lfloor \sqrt{n}\rfloor$, $d=0.1n$, Hurst parameters $H=0.6, 0.7$, and cross-covariance ${\operatorname{Cov}}(X_i, Y_{j})=r {\operatorname{Cov}}(X_i, X_{j})$, $1\leq i, j\leq n$, with $r=0$ and $r=0.25$. The level of significance equals 5%.

Theorems & Definitions (39)

• Lemma 2.1
• Theorem 2.2
• Remark 2.1
• Corollary 2.1
• Theorem 2.3
• Theorem 2.4
• Remark 2.2
• Theorem 2.5
• Theorem 2.6
• Remark 2.3