Nonuniform Berry-Esseen bounds for martingales with applications to statistical estimation

TL;DR

This paper derives nonuniform Berry-Esseen bounds for martingales under the conditional Bernstein condition, providing insights into large deviations and statistical applications like linear regressions and self-normalized deviations.

Contribution

It introduces new nonuniform Berry-Esseen bounds for martingales under the conditional Bernstein condition, with implications for large deviations and statistical estimation.

Findings

Bounds imply Cramér type large deviations for moderate x

Bounds exhibit exponential decay similar to de la Peña's inequality for large x

Applications include linear regression and self-normalized large deviations

Abstract

We establish nonuniform Berry-Esseen bounds for martingales under the conditional Bernstein condition. These bounds imply Cram\'er type large deviations for moderate $x$'s, and are of exponential decay rate as de la Pe\~na's inequality when $x\rightarrow \infty$. Statistical applications associated with linear regressions and self-normalized large deviations are also provided.