On the Bennett-Hoeffding inequality

TL;DR

This paper refines the Bennett-Hoeffding inequality by incorporating truncated third moments and a broader class of functions, resulting in bounds with improved optimality properties and extensions to martingales.

Contribution

It introduces a refined Bennett-Hoeffding bound using generalized moment functions and truncated third moments, enhancing the inequality's accuracy and applicability.

Findings

Refined bounds with truncated third moments.

Improved bounds using generalized moment functions.

Extensions to martingale maximal functions.

Abstract

The well-known Bennett-Hoeffding bound for sums of independent random variables is refined, by taking into account truncated third moments, and at that also improved by using, instead of the class of all increasing exponential functions, the much larger class of all generalized moment functions f such that f and f" are increasing and convex. It is shown that the resulting bounds have certain optimality properties. Comparisons with related known bounds are given. The results can be extended in a standard manner to (the maximal functions of) (super)martingales.