Asymptotic Behaviour of the Empirical Distance Covariance for Dependent Data

Abstract

We give two asymptotic results for the distance covariance on separable metric spaces without any iid assumptions. In particular, we show the almost sure convergence of the empirical distance covariance under ergodicity for any measure with finite first moments. We further give a result concerning the asymptotic distribution of the empirical distance covariance under the assumption of absolute regularity 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.