On the rate of convergence in Wasserstein distance of the empirical measure

Abstract

Let μN be the empirical measure associated to a N-sample of a given probability distribution μ on Rd. We are interested in the rate of convergence of μN to μ, when measured in the Wasserstein distance of order p>0. We provide some satisfying non-asymptotic Lp-bounds and concentration inequalities, for any values of p>0 and d≥1. We extend also the non asymptotic Lp-bounds to stationary ρ-mixing sequences, Markov chains, and to some interacting particle systems.