Simple bounds for convergence of empirical and occupation measures in 1-Wasserstein distance

We study the problem of non-asymptotic deviations between a reference measure mu and its empirical version L-n, in the 1-Wasserstein metric, under the standing assumption that mu satisfies a transport-entropy inequality. We extend some results of F. Bolley, A. Guillin and C. Villani [8] with simple proofs. Our methods are based on concentration inequalities and extend to the general setting of measures on a Polish space. Deviation bounds for the occupation measure of a contracting Markov chain in W-1 distance are also given. Throughout the text, several examples are worked out, including the cases of Gaussian measures on separable Banach spaces, and laws of diffusion processe.