On the links between Stein transforms and concentration inequalities for dependent random variables

TL;DR

This paper investigates the relationship between Stein's method transforms and concentration inequalities, establishing new bounds for dependent variables and linking stochastic domination to sub-Gaussian concentration.

Contribution

It introduces a new stochastic order connecting Stein transforms to concentration, and extends concentration bounds to dependent, light-tailed variables.

Findings

Zero bias transform domination implies sub-Gaussian concentration

Recovered McDiarmid inequality for bounded differences

Derived new bounds for dependent light-tailed variables

Abstract

In this paper, we explore some links between transforms derived by Stein's method and concentration inequalities. In particular, we show that the stochastic domination of the zero bias transform of a random variable is equivalent to sub-Gaussian concentration. For this purpose a new stochastic order is considered. In a second time, we study the case of functions of slightly dependent light-tailed random variables. We are able to recover the famous McDiarmid type of concentration inequality for functions with the bounded difference property. Additionally, we obtain new concentration bounds when we authorize a light dependence between the random variables. Finally, we give a analogous result for another type of Stein's transform, the so-called size bias transform.