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

Original · EN

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.