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

Let be the empirical measure associated to a -sample of a given probability distribution on . We are interested in the rate of convergence of to , when measured in the Wasserstein distance of order . We provide some satisfying non-asymptotic -bounds and concentration inequalities, for any values of and . We extend also the non asymptotic -bounds to stationary -mixing sequences, Markov chains, and to some interacting particle systems.