Self-normalized Cramér type moderate deviations for martingales
Résumé
Let $(X _i,\mathcal{F}_i)_{i\geq1}$ be a sequence of martingale differences. Set $S_n=\sum_{i=1}^n X_i $ and $[ S]_n=\sum_{i=1}^n X_i^2.$ 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, Zhao and Wang 2003] for independent random variables.