On the Missing Factor in Some Concentration Inequalities for Martingales

TL;DR

This paper improves concentration inequalities for martingales with bounded increments, recovering a missing factor in Freedman-style bounds and providing refinements for functions of independent variables using advanced techniques.

Contribution

It introduces near-optimal concentration inequalities for martingales, addressing a missing factor in previous bounds and refining inequalities for independent functions.

Findings

Recovered the missing factor in Freedman-style inequalities

Provided near-optimal bounds for martingales with bounded increments

Refined concentration inequalities for functions of independent variables

Abstract

In this note, we improve some concentration inequalities for martingales with bounded increments. These results recover the missing factor in Freedman-style inequalities and are near optimal. We also provide minor refinements of concentration inequalities for functions of independent random variables. These proofs use techniques from the works of Bentkus and Pinelis.