Berry-Esseen bounds for self-normalized martingales

TL;DR

This paper derives a Berry-Esseen bound for self-normalized martingales with finite moments, matching classical bounds for standardized martingales, and demonstrates its optimality with applications to Student's t-statistic and autoregressive processes.

Contribution

It introduces a Berry-Esseen bound specifically for self-normalized martingales, extending classical results to this broader context.

Findings

Bound matches classical Berry-Esseen bounds for standardized martingales

Example shows the bound's optimality

Applications to Student's t-statistic and autoregressive processes

Abstract

A Berry-Esseen bound is obtained for self-normalized martingales under the assumption of finite moments. The bound coincides with the classical Berry-Esseen bound for standardized martingales. An example is given to show the optimality of the bound. Applications to Student's statistic and autoregressive process are also discussed.