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.