Detecting independence of random vectors: generalized distance covariance and Gaussian covariance
Volume 5, Issue 3 (2018), pp. 353–383
Pub. online: 19 September 2018
Type: Research Article
Open Access
Open Access
Received
29 June 2018
29 June 2018
Revised
23 August 2018
23 August 2018
Accepted
30 August 2018
30 August 2018
Published
19 September 2018
19 September 2018
Abstract
Distance covariance is a quantity to measure the dependence of two random vectors. We show that the original concept introduced and developed by Székely, Rizzo and Bakirov can be embedded into a more general framework based on symmetric Lévy measures and the corresponding real-valued continuous negative definite functions. The Lévy 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.
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