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Self-normalized Cramer type moderate deviations for martingales (1712.04756v2)
Published 13 Dec 2017 in math.PR
Abstract: Let $(\xi_i,\mathcal{F}i){i\geq1}$ be a sequence of martingale differences. Set $S_n=\sum_{i=1}n\xi_i $ and $[ S]n=\sum{i=1}n \xi_i2.$ 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, Shao and Wang, 2003] for independent random variables.