Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Characterization of a class of weak transport-entropy inequalities on the line

Nathael Gozlan, Cyril Roberto, Paul-Marie Samson, Yan Shu, and Prasad Tetali

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We study an weak transport cost related to the notion of convex order between probability measures. On the real line, we show that this weak transport cost is reached for a coupling that does not depend on the underlying cost function. As an application, we give a necessary and sufficient condition for weak transport-entropy inequalities (related to concentration of convex/concave functions) to hold on the line. In particular, we obtain a weak transport-entropy form of the convex Poincaré inequality in dimension one.


Dans cet article, nous étudions une nouvelle famille de coûts de transport optimaux faibles en lien avec la notion d’ordre convexe pour les mesures de probabilité. Nous montrons, en dimension un, que le couplage optimal ne dépend pas de la fonction de coût choisie. Nous utilisons ensuite ce résultat pour établir une condition nécessaire et suffisante pour les inégalités de transport-entropie associées à ces coûts de transport faibles. En particulier, nous obtenons une forme transport équivalente de l’inégalité de Poincaré restreinte aux fonctions convexes sur la droite.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 3 (2018), 1667-1693.

Received: 13 January 2016
Revised: 6 June 2017
Accepted: 4 July 2017
First available in Project Euclid: 11 July 2018

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Zentralblatt MATH identifier

Primary: 60E15: Inequalities; stochastic orderings 32F32: Analytical consequences of geometric convexity (vanishing theorems, etc.) 26D10: Inequalities involving derivatives and differential and integral operators

Transport inequalities Concentration of measure Majorization


Gozlan, Nathael; Roberto, Cyril; Samson, Paul-Marie; Shu, Yan; Tetali, Prasad. Characterization of a class of weak transport-entropy inequalities on the line. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 3, 1667--1693. doi:10.1214/17-AIHP851.

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