Detecting independence of random vectors: generalized distance covariance and Gaussian covariance

TL;DR

This paper generalizes the concept of distance covariance using Lévy measures, enabling more flexible dependence measurement between random vectors with fewer restrictions and paving the way for multivariance analysis.

Contribution

It introduces a generalized framework for distance covariance based on Lévy measures, broadening its applicability and theoretical properties.

Findings

Preserves essential properties of distance covariance

Allows use of non-Euclidean distances like Minkowski distance

Serves as foundation for multivariance dependence measures

Abstract

Distance covariance is a quantity to measure the dependence of two random vectors. We show that the original concept introduced and developed by Sz\'{e}kely, Rizzo and Bakirov can be embedded into a more general framework based on symmetric L\'{e}vy measures and the corresponding real-valued continuous negative definite functions. The L\'{e}vy measures replace the weight functions used in the original definition of distance covariance. All essential properties of distance covariance are preserved in this new framework. From a practical point of view this allows less restrictive moment conditions on the underlying random variables and one can use other distance functions than Euclidean distance, e.g. Minkowski distance. Most importantly, it serves as the basic building block for distance multivariance, a quantity to measure and estimate dependence of multiple random vectors, which is introduced in a follow-up paper [Distance Multivariance: New dependence measures for random vectors (submitted). Revised version of arXiv: 1711.07775v1] to the present article.