# A Tight Bound of Tail Probabilities for a Discrete-time Martingale with Uniformly Bounded Jumps

## TL;DR

This paper derives a precise, tight upper bound for tail probabilities of discrete-time martingales with bounded jumps, improving upon the classical Azuma-Hoeffding inequality by providing explicit expressions and analyzing their asymptotic differences.

## Contribution

The paper introduces an explicit tight upper bound for tail probabilities of bounded-jump martingales, surpassing the Azuma-Hoeffding inequality in tightness and asymptotic accuracy.

## Key findings

- Derived an explicit tight upper bound for tail probabilities.
- Showed the new bound differs asymptotically from Azuma-Hoeffding.
- Validated the bound's tightness and practical relevance.

## Abstract

We investigate the properties of a discrete-time martingale $\{X_m\}_{m\in \mathbb Z_{\geq 0}}$, where all differences between adjacent random variables are limited to be not more than a constant as a promise. In this situation, it is known that the Azuma-Hoeffding inequality holds, which gives an upper bound of a probability for exceptional events. The inequality gives a simple form of the upper bound, and it has been utilized for many investigations. However, the inequality is not tight. We give an explicit expression of a tight upper bound, and we show that it and the bound obtained from the Azuma-Hoeffding inequality have different asymptotic behaviors.