• Bernoulli
  • Volume 25, Number 4A (2019), 2793-2823.

Self-normalized Cramér type moderate deviations for martingales

Xiequan Fan, Ion Grama, Quansheng Liu, and Qi-Man Shao

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Let $(X_{i},\mathcal{F}_{i})_{i\geq 1}$ 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 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 Jing, Shao and Wang (Ann. Probab. 31 (2003) 2167–2215) for independent random variables.

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Bernoulli, Volume 25, Number 4A (2019), 2793-2823.

Received: February 2018
Revised: June 2018
First available in Project Euclid: 13 September 2019

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Cramér’s moderate deviations martingales self-normalized sequences


Fan, Xiequan; Grama, Ion; Liu, Quansheng; Shao, Qi-Man. Self-normalized Cramér type moderate deviations for martingales. Bernoulli 25 (2019), no. 4A, 2793--2823. doi:10.3150/18-BEJ1071.

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Supplemental materials

  • Supplement to “Self-normalized Cramér type moderate deviations for martingales”. The supplement gives the detailed proofs of Propositions 3.1 and 3.2.