A new characterization of quadratic transportation-information inequalities

TL;DR

This paper introduces a new characterization of quadratic transportation-information inequalities, establishing their equivalence with Lyapunov conditions, their stability under perturbations, and their connection to Talagrand's inequality.

Contribution

It provides a novel equivalence between $ ext{W}_2I$ and Lyapunov conditions, and demonstrates the stability of $ ext{W}_2I$ under bounded perturbations, expanding the theoretical understanding of these inequalities.

Findings

Proves the equivalence of $ ext{W}_2I$ and Lyapunov conditions.

Shows stability of $ ext{W}_2I$ under bounded perturbations.

Derives $ ext{W}_2H$ from a restricted $ ext{W}_2I$.

Abstract

It is known that a quadratic transportation-information inequality $\mathrm{W_2I}$ interpolates between the Talagrand's inequality $\mathrm{W_2H}$ and the log-Sobolev inequality (LSI for short). The aim of the present paper is threefold: (1) To prove the equivalence of $\mathrm{W_2I}$ and the Lyapunov condition, which gives a new characterization inspired by Cattiaux-Guillin-Wu [8]. (2) To prove the stability of $\mathrm{W_2I}$ under bounded perturbations, which gives a transference principle in the sense of Holley-Stroock. (3) To prove $\mathrm{W_2H}$ through a restricted $\mathrm{W_2I}$, which gives a counterpart of the restricted LSI presented by Gozlan-Roberto-Samson [15].