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arXiv · 1905.06003

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

Go Kato
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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.

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Go Kato. 2019-05-15. A Tight Bound of Tail Probabilities for a Discrete-time Martingale with Uniformly Bounded Jumps. https://arxiv.org/abs/1905.06003

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