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

Abstract

We study the problem of non-asymptotic deviations between a reference measure μ and its empirical version Ln, in the 1-Wasserstein metric, under the standing assumption that μ 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 W1 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. © 2011 Applied Probability Trust.