Asymptotic Behaviour of the Empirical Distance Covariance for Dependent Data

Marius Kroll

doi:10.1007/s10959-021-01073-w

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.

Highlights

In [12], Lyons introduced the concept of distance covariance for separable metric spaces, generalising the work done by Székely et al [17]
In this very general case, the distance covariance of a measure θ with marginal distributions μ on X and ν on Y is defined as dcov(θ ) := δθ (z, z ) dθ 2(z, z )
It is of strong negative type if it is of negative type and D(μ1 − μ2) = 0 if and only if μ1 = μ2 for all probability measures μ1, μ2 with finite first moments

Summary

Introduction

In [12], Lyons introduced the concept of distance covariance for separable metric spaces, generalising the work done by Székely et al [17]. Acknowledged this in [13] (iii), showing that Proposition 2.6 as written in [12] is still correct in the case of spaces of negative type, but leaving the question of whether finite first moments are sufficient in the general case of separable metric spaces unanswered. This problem was solved in [9], where the almost sure convergence is shown in the case of iid samples. Definitions are taken from [4], where many properties of α-mixing and absolutely regular processes are established

Results for Metric Spaces

Generalisation to Pseudometric Spaces

Further Work

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