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Article Dans Une Revue Bernoulli Année : 2019

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.

Dates et versions

hal-02019056 , version 1 (14-02-2019)

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Xiequan Fan, Ion Grama, Quansheng Liu, Qi-Man Shao. Self-normalized Cramér type moderate deviations for martingales. Bernoulli, In press, ⟨10.3150/18-BEJ1071⟩. ⟨hal-02019056⟩
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