Electronic Communications in Probability

Large deviation exponential inequalities for supermartingales

Xiequan Fan, Ion Grama, and Quansheng Liu

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Let $(X_{i}, \mathcal{F}_{i})_{i\geq 1}$ be a sequence of supermartingale differences and let $S_k=\sum_{i=1}^k X_i$. We give an exponential moment condition under which $\mathbb{P}( \max_{1\leq k \leq n} S_k \geq n)=O(\exp\{-C_1 n^{\alpha}\}),$ $n\rightarrow \infty, $ where $\alpha \in $ is given and $C_{1}>0$ is a constant. We also show that the power $\alpha$ is optimal under the given moment condition.

Article information

Electron. Commun. Probab., Volume 17 (2012), paper no. 59, 8 pp.

Accepted: 12 December 2012
First available in Project Euclid: 7 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F10: Large deviations
Secondary: 60G42: Martingales with discrete parameter 60E15: Inequalities; stochastic orderings

Large deviation martingales exponential inequality Bernstein type inequality

This work is licensed under a Creative Commons Attribution 3.0 License.


Fan, Xiequan; Grama, Ion; Liu, Quansheng. Large deviation exponential inequalities for supermartingales. Electron. Commun. Probab. 17 (2012), paper no. 59, 8 pp. doi:10.1214/ECP.v17-2318. https://projecteuclid.org/euclid.ecp/1465263192

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