Modified log-Sobolev inequalities for convex functions on the real line. Sufficient conditions

TL;DR

This paper establishes a mild sufficient condition for probability measures on the real line to satisfy a modified log-Sobolev inequality for convex functions, leading to dimension-free concentration results and linking to weak transport-entropy inequalities.

Contribution

It introduces a new sufficient condition that interpolates between classical and Bobkov-Ledoux inequalities, expanding understanding of concentration for convex functions.

Findings

Dimension-free two-level concentration for convex functions

Link between modified log-Sobolev and weak transport-entropy inequalities

Sufficient conditions for measures on the real line

Abstract

We provide a mild sufficient condition for a probability measure on the real line to satisfy a modified log-Sobolev inequality for convex functions, interpolating between the classical log-Sobolev inequality and a Bobkov-Ledoux type inequality. As a consequence we obtain dimension-free two-level concentration results for convex function of independent random variables with sufficiently regular tail decay. We also provide a link between modified log-Sobolev inequalities for convex functions and weak transport-entropy inequalities, complementing recent work by Gozlan, Roberto, Samson, and Tetali.