Self-normalized Cramer type moderate deviations for martingales

Abstract

Let $({\xi }_i,{\mathcal{F}}_i{)}_{i\ge 1}$ 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 Cramér type moderate deviation expansion for $\mathbf{P}(S_n/\sqrt{[S{]}_n}\ge x)$ as $n\to +\infty .$ Our results partly extend the earlier work of [Jing, Shao and Wang, 2003] for independent random variables.