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.

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.

DOI : 10.1214/12-AIHP494
Classification : 60E15, 52A40, 62H05
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] F. Barthe. 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] C. Borell. A Gaussian correlation inequality for certain bodies in 𝐑 n . Math. Ann. 256 (4) (1981) 569-573. | MR | Zbl

[3] G. Harge. A particular case of correlation inequality for the Gaussian measure. Ann. Probab. 27 (4) (1999) 1939-1951. | MR | Zbl

[4] C. G. Khatri. On certain inequalities for normal distributions and their applications to simultaneous confidence bounds. Ann. Math. Statist. 38 (1967) 1853-1867. | MR | Zbl

[5] A. V. Kolesnikov. 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] L. D. Pitt. A Gaussian correlation inequality for symmetric convex sets. Ann. Probab. 5 (3) (1977) 470-474. | MR | Zbl

[7] G. Schechtman, T. Schlumprecht and J. Zinn. On the Gaussian measure of the intersection. Ann. Probab. 26 (1) (1998) 346-357. | MR | Zbl

[8] Z. Šidák. Rectangular confidence regions for the means of multivariate normal distributions. J. Amer. Statist. Assoc. 62 (1967) 626-633. | MR | Zbl

Cité par Sources :