FUNCTIONAL INEQUALITIES FOR TWO-LEVEL CONCENTRATION

Franck Barthe (1) , Michał Strzelecki

1 IMT - Institut de Mathématiques de Toulouse UMR5219

Probability measures satisfying a Poincaré inequality are known to enjoy a dimension free concentration inequality with exponential rate. A celebrated result of Bobkov and Ledoux shows that a Poincaré inequality automatically implies a modified logarithmic Sobolev inequality. As a consequence the Poincaré inequality ensures a stronger dimension free concentration property , known as two-level concentration. We show that a similar phenomenon occurs for the Latała-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.

Domaines: Analyse classique [math.CA]; Probabilités [math.PR]

Franck Barthe, Michał Strzelecki. FUNCTIONAL INEQUALITIES FOR TWO-LEVEL CONCENTRATION. 2019. ⟨hal-02281782v2⟩