Local tail bounds for functions of independent random variables

TL;DR

This paper demonstrates that functions with bounded differences on discrete hypercubes exhibit strong local tail bounds, extending Talagrand's variance inequality to a broader class of functions and distributions.

Contribution

It extends Talagrand's variance inequality to functions on \\{0,1,...,r-1\\}^n and establishes local tail bounds for functions with bounded differences and self-bounding properties.

Findings

Functions with bounded differences have local sub-Gaussian tail bounds.

Extension of Talagrand's inequality to functions on \\{0,1,...,r-1\\}^n.

Derived local subexponential bounds for self-bounding functions.

Abstract

It is shown that functions defined on $\{0,1,...,r-1\}^n$ satisfying certain conditions of bounded differences that guarantee sub-Gaussian tail behavior also satisfy a much stronger ``local'' sub-Gaussian property. For self-bounding and configuration functions we derive analogous locally subexponential behavior. The key tool is Talagrand's [Ann. Probab. 22 (1994) 1576--1587] variance inequality for functions defined on the binary hypercube which we extend to functions of uniformly distributed random variables defined on $\{0,1,...,r-1\}^n$ for $r\ge2$.