Optimal re-centering bounds, with applications to Rosenthal-type concentration of measure inequalities

TL;DR

This paper derives optimal bounds for re-centering inequalities involving functions of random variables and applies these results to establish Rosenthal-type concentration inequalities for Lipschitz functions on product spaces.

Contribution

It provides the exact best constants in re-centering inequalities and extends these results to obtain new concentration bounds of Rosenthal type.

Findings

Determined the optimal constants c_f for the inequality E f(X - E X) ≤ c_f E f(X).

Analyzed properties of c_f when f=|.|^p.

Applied these bounds to derive concentration inequalities for Lipschitz functions.

Abstract

For any nonnegative Borel-measurable function f such that f(x)=0 if and only if x=0, the best constant c_f in the inequality E f(X-E X) \leq c_f E f(X) for all random variables X with a finite mean is obtained. Properties of the constant c_f in the case when f=|.|^p are studied. Applications to concentration of measure in the form of Rosenthal-type bounds on the moments of separately Lipschitz functions on product spaces are given.