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.
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},
pages = {1--14},
year = {2014},
publisher = {Gauthier-Villars},
volume = {50},
number = {1},
doi = {10.1214/12-AIHP494},
mrnumber = {3161519},
zbl = {1288.60024},
language = {en},
url = {https://www.numdam.org/articles/10.1214/12-AIHP494/}
}
TY - JOUR AU - Figalli, A. AU - Maggi, F. AU - Pratelli, A. TI - A geometric approach to correlation inequalities in the plane JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2014 SP - 1 EP - 14 VL - 50 IS - 1 PB - Gauthier-Villars UR - https://www.numdam.org/articles/10.1214/12-AIHP494/ DO - 10.1214/12-AIHP494 LA - en ID - AIHPB_2014__50_1_1_0 ER -
%0 Journal Article %A Figalli, A. %A Maggi, F. %A Pratelli, A. %T A geometric approach to correlation inequalities in the plane %J Annales de l'I.H.P. Probabilités et statistiques %D 2014 %P 1-14 %V 50 %N 1 %I Gauthier-Villars %U https://www.numdam.org/articles/10.1214/12-AIHP494/ %R 10.1214/12-AIHP494 %G en %F 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, Tome 50 (2014) no. 1, pp. 1-14. doi: 10.1214/12-AIHP494
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