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Self-normalized Cramér type moderate deviations for martingales. (English) Zbl 1428.62424
Summary: Let $$(X_i,\mathcal{F}_i)_{i\geq 1}$$ be a sequence of martingale differences. Set $$S_n=\sum_{i=1}^nX_i$$ and $$[S]_n=\sum_{i=1}^nX_i^2$$. We prove a Cramér 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 B.-Y. Jing et al. [Ann. Probab. 31, No. 4, 2167–2215 (2003; Zbl 1051.60031)] for independent random variables.

##### MSC:
 60G42 Martingales with discrete parameter 60F10 Large deviations
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##### References:
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