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

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