On McDiarmid's inequality for Hamming distance

TL;DR

This paper enhances McDiarmid's inequality for Hamming distance, providing tighter bounds and extensions to dependent variables, thereby improving convergence rate estimates for Lipschitz functions of independent and dependent random variables.

Contribution

The authors refine the rate function of McDiarmid's inequality for Hamming distance and extend it to dependent variables, offering more precise convergence bounds.

Findings

Improved rate function for McDiarmid's inequality with Hamming distance

Refined convergence rate around the median for Lipschitz functions

Extended inequalities to nonnegative functionals of dependent variables

Abstract

We improve the rate function of McDiarmid's inequality for Hamming distance. In particular, applying our result to the separately Lipschitz functions of independent random variables, we also refine the convergence rate function of McDiarmid's inequality around a median. Moreover, a non-uniform bound for the distance between the medians and the mean is also given. We also give some extensions of McDiarmid's inequalities to the case of nonnegative functionals of dependent random variables.