Rates of convergence in the distances of Kolmogorov and Wasserstein for standardized martingales

TL;DR

This paper establishes convergence rates in Kolmogorov and Wasserstein distances for standardized martingales, extending Berry-Esseen bounds and demonstrating optimality of rates using Stein's method.

Contribution

It provides new Berry-Esseen bounds for martingales and confirms the optimality of Wasserstein convergence rates with Stein's method.

Findings

Exact Berry-Esseen bounds for martingales in Kolmogorov distance.

Optimal convergence rates in Wasserstein distance using Stein's method.

Recovery of classical rates for sums of i.i.d. variables.

Abstract

We give some rates of convergence in the distances of Kolmogorov and Wasserstein for standardized martingales with differences having finite variances. For the Kolmogorov distances, we present some exact Berry-Esseen bounds for martingales, which generalizes some Berry-Esseen bounds due to Bolthausen. For the Wasserstein distance, with Stein's method and Lindeberg's telescoping sum argument, the rates of convergence in martingale central limit theorems recover the classical rates for sums of i.i.d.\ random variables, and therefore they are believed to be optimal.