Bernoulli

  • 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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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.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.bj/1568362043

Digital Object Identifier
doi:10.3150/18-BEJ1071

Keywords
Cramér’s moderate deviations martingales self-normalized sequences

Citation

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. https://projecteuclid.org/euclid.bj/1568362043


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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.