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

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.