Asymptotic Behaviour of the Empirical Distance Covariance for Dependent Data

TL;DR

This paper establishes the asymptotic properties of empirical distance covariance for dependent data, including convergence and distribution results, without requiring iid assumptions, thus broadening its applicability in statistical dependence testing.

Contribution

It provides the first asymptotic analysis of empirical distance covariance for dependent, non-iid data, including convergence and distribution theorems under stationarity and mixing conditions.

Findings

Almost sure convergence for stationary ergodic processes.

Asymptotic distribution under absolute regularity.

Extension to pseudometric spaces.

Abstract

We give two asymptotic results for the empirical distance covariance on separable metric spaces without any iid assumption on the samples. In particular, we show the almost sure convergence of the empirical distance covariance for any measure with finite first moments, provided that the samples form a strictly stationary and ergodic process. We further give a result concerning the asymptotic distribution of the empirical distance covariance under the assumption of absolute regularity of the samples and extend these results to certain types of pseudometric spaces. In the process, we derive a general theorem concerning the asymptotic distribution of degenerate V-statistics of order 2 under a strong mixing condition.