Large deviation exponential inequalities for supermartingales

Abstract

Let $(X_i,{\mathcal{F}}_i{)}_{i\ge 1}$ be a sequence of supermartingale differences and let $S_k={\sum }_{i=1}^k{X}_i$. We give an exponential moment condition under which $P({\max }_{1\le k\le n}{S}_k\ge n)=O(\exp \{-C_1{n}^{\alpha }\}),$ $n\to \infty ,$ where $\alpha \in (0,1)$ is given and $C_1>0$ is a constant. We also show that the power $\alpha$ is optimal under the given condition. In particular, when $\alpha =1/3$, we recover an inequality of Lesigne and Volný.

9 pages