Nonuniform Berry-Esseen bound for self-normalized martingales

TL;DR

This paper establishes a nonuniform Berry-Esseen bound for self-normalized martingales, connecting previous results and providing a more precise convergence rate for these stochastic processes.

Contribution

It introduces a new nonuniform Berry-Esseen bound for self-normalized martingales, bridging existing gaps between prior bounds for standardized and self-normalized cases.

Findings

Bound coincides with Haeusler and Joos (1988) for standardized martingales

Provides a Berry-Esseen bound for self-normalized martingales

Bridges the gap between earlier results of Haeusler (1988) and Fan and Shao (2018)

Abstract

We give a nonuniform Berry-Esseen bound for self-normalized martingales, which bridges the gap between the result of Haeusler (1988) and Fan and Shao (2018). The bound coincides with the nonuniform Berry-Esseen bound of Haeusler and Joos (1988) for standardized martingales. As a consequence, a Berry-Esseen bound is obtained.