A geometric approach to correlation inequalities in the plane
Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 1, p. 1-14

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.

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.

DOI : https://doi.org/10.1214/12-AIHP494
Classification:  60E15,  52A40,  62H05
Keywords: correlation inequalities, gaussian correlation conjecture, radially symmetric measures
@article{AIHPB_2014__50_1_1_0,
     author = {Figalli, A. and Maggi, F. and Pratelli, A.},
     title = {A geometric approach to correlation inequalities in the plane},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {1},
     year = {2014},
     pages = {1-14},
     doi = {10.1214/12-AIHP494},
     zbl = {1288.60024},
     mrnumber = {3161519},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2014__50_1_1_0}
}
Figalli, A.; Maggi, F.; Pratelli, A. A geometric approach to correlation inequalities in the plane. Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 1, pp. 1-14. doi : 10.1214/12-AIHP494. http://www.numdam.org/item/AIHPB_2014__50_1_1_0/

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