Behavior of the Wasserstein distance between the empirical and the marginal distributions of stationary $\alpha$-dependent sequences

Abstract

We study the Wasserstein distance of order 1 between the empirical distribution and the marginal distribution of stationary $\alpha$-dependent sequences. We prove some moments inequalities of order $p$ for any $p\geq1$, and we give some conditions under which the central limit theorem holds. We apply our results to unbounded functions of expanding maps of the interval with a neutral fixed point at zero. The moment inequalities for the Wasserstein distance are similar to the well-known von Bahr–Esseen or Rosenthal bounds for partial sums, and seem to be new even in the case of independent and identically distributed random variables.