New-Type Hoeffding's Inequalities and Application in Tail Bounds

TL;DR

This paper introduces a new class of Hoeffding's inequalities that incorporate higher-order moments of random variables, leading to improved tail bounds and potential applications in signal processing and related fields.

Contribution

It presents a novel type of Hoeffding's inequalities considering higher-order moments, enhancing tail bound accuracy over existing first-moment-based results.

Findings

Improved tail bounds using higher-order moments.

Enhanced Hoeffding-Azuma inequalities for martingales.

Potential for broader applications in signal and information processing.

Abstract

It is well known that Hoeffding's inequality has a lot of applications in the signal and information processing fields. How to improve Hoeffding's inequality and find the refinements of its applications have always attracted much attentions. An improvement of Hoeffding inequality was recently given by Hertz \cite{r1}. Eventhough such an improvement is not so big, it still can be used to update many known results with original Hoeffding's inequality, especially for Hoeffding-Azuma inequality for martingales. However, the results in original Hoeffding's inequality and its refinement one by Hertz only considered the first order moment of random variables. In this paper, we present a new type of Hoeffding's inequalities, where the high order moments of random variables are taken into account. It can get some considerable improvements in the tail bounds evaluation compared with the known results. It is expected that the developed new type Hoeffding's inequalities could get more interesting applications in some related fields that use Hoeffding's results.