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

TL;DR

This paper investigates the behavior of the Wasserstein distance between empirical and marginal distributions in stationary $eta$-dependent sequences, providing moment inequalities and CLT conditions, with applications to expanding maps.

Contribution

It introduces new moment inequalities for the Wasserstein distance in dependent sequences and establishes CLT conditions, extending known results to $eta$-dependent processes.

Findings

Derived moment inequalities similar to von Bahr-Esseen bounds.

Established CLT conditions for Wasserstein distance in dependent sequences.

Applied results to expanding maps with neutral fixed points.

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 $\ge$ 1, 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.