Mean rates of convergence of empirical measures in the Wasserstein metric

Abstract

An upper bound is given for the mean square Wasserstein distance between the empirical measure of a sequence of i.i.d. random vectors and the common probability law of the sequence. The same result holds for an infinite exchangeable sequence and its directing measure. Similarly, for an i.i.d. sequence of stochastic processes, an upper bound is obtained for the mean square of the maximum, over 0 ⩽ t ⩽ T, of the Wasserstein distance between the empirical measure of the sequence at time t and the common marginal law at t. These upper bounds are derived under weak assumptions and are not very far from the known rate of convergence pertaining to an i.i.d. sequence of uniform random vectors on the unit cube. Our approach, however, allows us to get results for arbitrary distributions under moment conditions and also gives results for processes. An application is given to so-called diffusions with jumps. Moment estimates for these processes are derived which may be of independent interest.