Functional inequalities for two-level concentration

TL;DR

This paper explores how certain functional inequalities, like Poincaré and Latala-Oleszkiewicz, imply strong, dimension-free concentration properties, and develops techniques to analyze measures with complex potentials.

Contribution

It extends the understanding of dimension-free concentration by linking Latala-Oleszkiewicz inequalities to two-level concentration and introduces new analytic methods for measures with wild potentials.

Findings

Poincaré inequality implies two-level concentration.

Latala-Oleszkiewicz inequalities also lead to dimension-free concentration.

Developed techniques for analyzing measures with complex potentials.

Abstract

Probability measures satisfying a Poincar{\'e} inequality are known to enjoy a dimension free concentration inequality with exponential rate. A celebrated result of Bobkov and Ledoux shows that a Poincar{\'e} inequality automatically implies a modified logarithmic Sobolev inequality. As a consequence the Poincar{\'e} inequality ensures a stronger dimension free concentration property , known as two-level concentration. We show that a similar phenomenon occurs for the Latala-Oleszkiewicz inequalities, which were devised to uncover dimension free concentration with rate between exponential and Gaussian. Motivated by the search for counterexamples to related questions, we also develop analytic techniques to study functional inequalities for probability measures on the line with wild potentials.