Large deviation exponential inequalities for supermartingales

TL;DR

This paper establishes exponential inequalities for the maximum of supermartingale sums under certain moment conditions, demonstrating the optimality of the decay rate and extending previous results for specific cases.

Contribution

It introduces a new exponential moment condition for supermartingales that yields sharp tail bounds with optimal decay rates.

Findings

Derived exponential decay bounds for supermartingale maxima

Proved the optimality of the decay rate exponent

Extended and unified previous inequalities for specific cases

Abstract

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 $P(\max_{1\leq k \leq n} S_k \geq n)=O(\exp\{-C_1 n^{\alpha}\}),$ $n\rightarrow \infty,$ 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 condition. In particular, when $\alpha=1/3$, we recover an inequality of Lesigne and Voln\'{y}.