Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

A characterization of a class of convex log-Sobolev inequalities on the real line

Yan Shu and Michał Strzelecki

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Abstract

We give a sufficient and necessary condition for a probability measure $\mu$ on the real line to satisfy the logarithmic Sobolev inequality for convex functions. The condition is expressed in terms of the unique left-continuous and non-decreasing map transporting the symmetric exponential measure onto $\mu$. The main tool in the proof is the theory of weak transport costs.

As a consequence, we obtain dimension-free concentration bounds for the lower and upper tails of convex functions of independent random variables which satisfy the convex log-Sobolev inequality.

Résumé

Nous proposons une condition nécessaire et suffisante pour qu’une mesure de probabilité $\mu$ sur la droite réelle satisfasse une condition de Sobolev logarithmique sur les fonctions convexes. Cette condition est exprimée en termes de l’unique plan de transport optimal croissant et continu à gauche entre la mesure exponentielle symétrique et la mesure $\mu$. L’outil principal vient de la théorie du transport faible.

Comme conséquence, nous obtenons un résultat de concentration adimensionnelle sur les estimées de queue de fonctions convexes de variables aléatoires indépendantes, lié à l’inégalité de Sobolev logarithmique convexe.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 4 (2018), 2075-2091.

Dates
Received: 16 February 2017
Revised: 23 May 2017
Accepted: 20 September 2017
First available in Project Euclid: 18 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1539849793

Digital Object Identifier
doi:10.1214/17-AIHP865

Mathematical Reviews number (MathSciNet)
MR3865667

Zentralblatt MATH identifier
06996559

Subjects
Primary: 60E15: Inequalities; stochastic orderings
Secondary: 26A51: Convexity, generalizations 26D10: Inequalities involving derivatives and differential and integral operators

Keywords
Concentration of measure Convex functions Log-Sobolev inequality Weak transport-entropy inequalities

Citation

Shu, Yan; Strzelecki, Michał. A characterization of a class of convex log-Sobolev inequalities on the real line. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 4, 2075--2091. doi:10.1214/17-AIHP865. https://projecteuclid.org/euclid.aihp/1539849793


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