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

Let \μN be the empirical measure associated to a N-sample of a given distribution \μ on ᵈ. We are interested in the rate convergence of \μN to \μ, 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 \ρ-mixing sequences, chains, and to some interacting particle systems.