Concentration of the empirical measure in Wasserstein distance: bounds involving the covering dimension

TL;DR

This paper develops concentration inequalities for empirical measures in Wasserstein distance, incorporating the covering dimension of the support, extending previous results to more general spaces beyond Euclidean dimensions.

Contribution

It introduces bounds involving the covering dimension for empirical measures in Wasserstein distance, generalizing Fournier and Guillin's inequalities to broader spaces.

Findings

Concentration inequalities depend on the covering dimension of the support.

Extension of Fournier and Guillin's inequalities to general Polish spaces.

Results applicable to empirical measures in various metric spaces.

Abstract

We give concentration inequalities in Wasserstein distance for the empirical measure of a sequence of independent and identically distributed random variables with values in a Polish space E. These inequalities involve the covering dimension of the support of the distribution of the variables. More precisely, we obtain a complete extension of the concentration inequalities of Fournier and Guillin [2015] in the case where E = R^d , in which the covering dimension replaces the dimension of the ambient space E.