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November 2019 Self-normalized Cramér type moderate deviations for martingales
Xiequan Fan, Ion Grama, Quansheng Liu, Qi-Man Shao
Bernoulli 25(4A): 2793-2823 (November 2019). DOI: 10.3150/18-BEJ1071

Abstract

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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Xiequan Fan. Ion Grama. Quansheng Liu. Qi-Man Shao. "Self-normalized Cramér type moderate deviations for martingales." Bernoulli 25 (4A) 2793 - 2823, November 2019. https://doi.org/10.3150/18-BEJ1071

Information

Received: 1 February 2018; Revised: 1 June 2018; Published: November 2019
First available in Project Euclid: 13 September 2019

zbMATH: 07110112
MathSciNet: MR4003565
Digital Object Identifier: 10.3150/18-BEJ1071

Rights: Copyright © 2019 Bernoulli Society for Mathematical Statistics and Probability

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Vol.25 • No. 4A • November 2019
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