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

Abstract

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