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.
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.
Mots clés : 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},
publisher = {Gauthier-Villars},
volume = {50},
number = {1},
year = {2014},
doi = {10.1214/12-AIHP494},
mrnumber = {3161519},
zbl = {1288.60024},
language = {en},
url = {http://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 - http://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 http://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. http://www.numdam.org/articles/10.1214/12-AIHP494/
[1] . Transportation techniques and Gaussian inequalities. In Optimal Transportation, Geometry, and Functional Inequalities. L. Ambrosio (Ed.) Edizioni della Scuola Normale Superiore di Pisa, 2010. | MR | Zbl
[2] . A Gaussian correlation inequality for certain bodies in . Math. Ann. 256 (4) (1981) 569-573. | MR | Zbl
[3] . A particular case of correlation inequality for the Gaussian measure. Ann. Probab. 27 (4) (1999) 1939-1951. | MR | Zbl
[4] . On certain inequalities for normal distributions and their applications to simultaneous confidence bounds. Ann. Math. Statist. 38 (1967) 1853-1867. | MR | Zbl
[5] . Geometrical properties of the diffusion semigroups and convex inequalities. Preprint 2006. Available at http://www.math.uni-bielefeld.de/~bibos/preprints/06-03-208.pdf.
[6] . A Gaussian correlation inequality for symmetric convex sets. Ann. Probab. 5 (3) (1977) 470-474. | MR | Zbl
[7] , and . On the Gaussian measure of the intersection. Ann. Probab. 26 (1) (1998) 346-357. | MR | Zbl
[8] . Rectangular confidence regions for the means of multivariate normal distributions. J. Amer. Statist. Assoc. 62 (1967) 626-633. | MR | Zbl
Cité par Sources :
