Distance covariance in metric spaces
Abstract
We extend the theory of distance (Brownian) covariance from Euclidean spaces, where it was introduced by Sz\'{e}kely, Rizzo and Bakirov, to general metric spaces. We show that for testing independence, it is necessary and sufficient that the metric space be of strong negative type. In particular, we show that this holds for separable Hilbert spaces, which answers a question of Kosorok. Instead of the manipulations of Fourier transforms used in the original work, we use elementary inequalities for metric spaces and embeddings in Hilbert spaces.
Keywords
Cite
@article{arxiv.1106.5758,
title = {Distance covariance in metric spaces},
author = {Russell Lyons},
journal= {arXiv preprint arXiv:1106.5758},
year = {2021}
}
Comments
Published at http://dx.doi.org/10.1214/12-AOP803 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org). This version has appended the two published errata as well