Behavior of the empirical wasserstein distance in rd under moment conditions

Abstract

We establish some deviation inequalities, moment bounds and almost sure results for the Wasserstein distance of order p ∈ [1, ∞) between the empirical measure of independent and identically distributed Rd-valued random variables and the common distribution of the variables. We only assume the existence of a (strong or weak) moment of order rp for some r > 1, and we discuss the optimality of the bounds.