Large deviation exponential inequalities for supermartingales

Xiequan Fan (1) , Ion Grama (1) , Quansheng Liu (1)

1 LMBA - Laboratoire de Mathématiques de Bretagne Atlantique

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 $\alpha \in (0, 1)$ is given and $C_{1}>0$ is a constant. We also show that the power $\alpha$ is optimal under the given moment condition.

Mots clés: Bernstein type inequality; Large deviation; martingales; exponential inequality

Domaines: Probabilités [math.PR]

Xiequan Fan, Ion Grama, Quansheng Liu. Large deviation exponential inequalities for supermartingales. Electronic Communications in Probability, 2012, 17 (59), pp.1-8. ⟨hal-00905554⟩