Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

A geometric approach to correlation inequalities in the plane

A. Figalli, F. Maggi, and A. Pratelli

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Abstract

By elementary geometric arguments, correlation inequalities for radially symmetric probability measures are proved in the plane. Precisely, it is shown that the correlation ratio for pairs of width-decreasing sets is minimized within the class of infinite strips. Since open convex sets which are symmetric with respect to the origin turn out to be width-decreasing sets, Pitt’s Gaussian correlation inequality (the two-dimensional case of the long-standing Gaussian correlation conjecture) is derived as a corollary, and it is in fact extended to a wide class of radially symmetric measures.

Résumé

En utilisant des arguments géométriques élémentaires, on démontre des inégalités de corrélation pour des mesures de probabilité à symétrie radiale. Plus précisément on montre que, parmi la famille des ensembles width-decreasing, le ratio de corrélation est minimisé par des bandes. Comme les ouverts convexes symétriques appartiennent à cette famille, on retrouve comme corollaire le résultat de Pitt sur la validité de la conjecture de corrélation gaussiennne en dimension 2, qui est étendue dans ce papier à une large classe de mesures à symétrie radiale.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 50, Number 1 (2014), 1-14.

Dates
First available in Project Euclid: 1 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1388545262

Digital Object Identifier
doi:10.1214/12-AIHP494

Mathematical Reviews number (MathSciNet)
MR3161519

Zentralblatt MATH identifier
1288.60024

Subjects
Primary: 60E15: Inequalities; stochastic orderings
Secondary: 52A40: Inequalities and extremum problems 62H05: Characterization and structure theory

Keywords
Correlation inequalities Gaussian correlation conjecture Radially symmetric measures

Citation

Figalli, A.; Maggi, F.; Pratelli, A. A geometric approach to correlation inequalities in the plane. Ann. Inst. H. Poincaré Probab. Statist. 50 (2014), no. 1, 1--14. doi:10.1214/12-AIHP494. https://projecteuclid.org/euclid.aihp/1388545262


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