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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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