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.
Citation
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
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