Large deviations for martingales with exponential condition

Details Large deviations for martingales with exponential condition Large deviations for martingales with exponential condition

2011-11-06

arxiv.org/abs/1111.1407v2

Mathematics

Probability

5 pages

Scientific paper

Let $(X_{i}, \mathcal{F}_{i})_{i=1,...,n}$ be a sequence of martingale differences and let $S_n=\sum_{i=1}^n X_i$. For any constant $\alpha \in (0, 1)$, we prove that if $E \exp{|\delta X_{i}|^{\frac{2\alpha}{1-\alpha}}} < \infty$ for some constant $\delta>0$ and all $i$, then $P(|S_n| > n)=O(\exp{-C_1 n^{\alpha}}),$ $n\rightarrow \infty, $ where $C_{1}>0$ is a constant. When $\alpha=1/3$ and $\delta=1$, this result reduces to that of Lesigne and Voln\'y (\textit{Stochastic Process. Appl.} 96 (2001) 143).

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