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

Let ${\mu }_N$ be the empirical measure associated to a $N$-sample of a given probability distribution $\mu$ on ${\mathbb{R}}^d$. We are interested in the rate of convergence of ${\mu }_N$ to $\mu$, when measured in the Wasserstein distance of order $p>0$. We provide some satisfying non-asymptotic $L^p$-bounds and concentration inequalities, for any values of $p>0$ and $d\ge 1$. We extend also the non asymptotic $L^p$-bounds to stationary $\rho$-mixing sequences, Markov chains, and to some interacting particle systems.