Wasserstein-p Bounds in the Central Limit Theorem Under Weak Dependence

TL;DR

This paper derives convergence rates in the Wasserstein-p distance for the central limit theorem applied to weakly dependent data, extending classical results to more general dependence structures.

Contribution

It introduces new bounds for Wasserstein-p distances in the CLT under weak dependence, using Stein's method adapted for dependent variables.

Findings

Established upper bounds for Wasserstein-p distance in dependent data.

Derived high-order local expansions of the Stein equation for dependent variables.

Provided asymptotically tight tail bounds for empirical averages of weakly dependent data.

Abstract

The central limit theorem is one of the most fundamental results in probability and has been successfully extended to locally dependent data and strongly-mixing random fields. In this paper, we establish its rate of convergence for transport distances, namely for arbitrary $p\ge1$ we obtain an upper bound for the Wasserstein-$p$ distance for locally dependent random variables and strongly mixing stationary random fields. Our proofs adapt the Stein dependency neighborhood method to the Wasserstein-$p$ distance and as a by-product we establish high-order local expansions of the Stein equation for dependent random variables. Finally, we demonstrate how our results can be used to obtain tail bounds that are asymptotically tight, and decrease polynomially fast, for the empirical average of weakly dependent random variables.