Self-normalized Cramer type moderate deviations for martingales

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 $[S{]}_n={\sum }_{i=1}^n{\xi }_i^2\mathrm{.}$ 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.