Self-normalized Cramer type moderate deviations for martingales

TL;DR

This paper establishes Cramér type moderate deviation results for self-normalized martingales, extending previous work on independent variables to the martingale setting, providing precise probability estimates for large deviations.

Contribution

It introduces a new moderate deviation expansion for martingales normalized by their quadratic variation, extending classical results to dependent structures.

Findings

Provides a Cramér type moderate deviation expansion for martingales.

Extends earlier results from independent variables to martingales.

Offers precise asymptotic probability estimates for large deviations.

Abstract

Let $(\xi_i,\mathcal{F}_i)_{i\geq1}$ be a sequence of martingale differences. Set $S_n=\sum_{i=1}^n\xi_i $ and $[ S]_n=\sum_{i=1}^n \xi_i^2.$ We prove a Cram\'er type moderate deviation expansion for $\mathbf{P}(S_n/\sqrt{[ S]_n} \geq x)$ as $n\to+\infty.$ Our results partly extend the earlier work of [Jing, Shao and Wang, 2003] for independent random variables.