Functional inequalities for heavy tails distributions and application to isoperimetry

TL;DR

This paper investigates functional inequalities for heavy-tailed probability measures, establishing new bounds and isoperimetric inequalities using Lyapunov functions and mass transportation, with applications to product and spherically symmetric measures.

Contribution

It introduces a unified approach to prove various functional inequalities for heavy-tailed measures, improving existing results and deriving optimal dimension dependence.

Findings

Established weak Poincaré and Cheeger inequalities for heavy tails

Derived optimal isoperimetric inequalities for product measures

Improved previous results for spherically symmetric measures

Abstract

This paper is devoted to the study of probability measures with heavy tails. Using the Lyapunov function approach we prove that such measures satisfy different kind of functional inequalities such as weak Poincar\'e and weak Cheeger, weighted Poincar\'e and weighted Cheeger inequalities and their dual forms. Proofs are short and we cover very large situations. For product measures on $\R^n$ we obtain the optimal dimension dependence using the mass transportation method. Then we derive (optimal) isoperimetric inequalities. Finally we deal with spherically symmetric measures. We recover and improve many previous results.