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

TL;DR

This paper extends optimal Wasserstein-p bounds in the CLT to locally dependent variables, including m-dependent fields and U-statistics, by developing new approximation techniques and Stein's method adaptations.

Contribution

It introduces novel bounds for the CLT under Wasserstein-p distance for dependent data, broadening the scope beyond independent variables.

Findings

Derived optimal convergence rates for dependent variables

Extended Stein's method to arbitrary orders for local dependence

Provided tail bounds demonstrating practical applicability

Abstract

The central limit theorem (CLT) is one of the most fundamental results in probability; and establishing its rate of convergence has been a key question since the 1940s. For independent random variables, a series of recent works established optimal error bounds under the Wasserstein-p distance (with p>=1). In this paper, we extend those results to locally dependent random variables, which include m-dependent random fields and U-statistics. Under conditions on the moments and the dependency neighborhoods, we derive optimal rates in the CLT for the Wasserstein-p distance. Our proofs rely on approximating the empirical average of dependent observations by the empirical average of i.i.d. random variables. To do so, we expand the Stein equation to arbitrary orders by adapting the Stein's dependency neighborhood method. Finally we illustrate the applicability of our results by obtaining efficient tail bounds.