# A geometric approach to correlation inequalities in the plane

A. Figalli; F. Maggi; A. Pratelli

Annales de l'I.H.P. Probabilités et statistiques (2014)

- Volume: 50, Issue: 1, page 1-14
- ISSN: 0246-0203

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topFigalli, A., Maggi, F., and Pratelli, A.. "A geometric approach to correlation inequalities in the plane." Annales de l'I.H.P. Probabilités et statistiques 50.1 (2014): 1-14. <http://eudml.org/doc/272063>.

@article{Figalli2014,

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.},

author = {Figalli, A., Maggi, F., Pratelli, A.},

journal = {Annales de l'I.H.P. Probabilités et statistiques},

keywords = {correlation inequalities; gaussian correlation conjecture; radially symmetric measures; Gaussian correlation conjecture},

language = {eng},

number = {1},

pages = {1-14},

publisher = {Gauthier-Villars},

title = {A geometric approach to correlation inequalities in the plane},

url = {http://eudml.org/doc/272063},

volume = {50},

year = {2014},

}

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

PB - Gauthier-Villars

VL - 50

IS - 1

SP - 1

EP - 14

AB - 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.

LA - eng

KW - correlation inequalities; gaussian correlation conjecture; radially symmetric measures; Gaussian correlation conjecture

UR - http://eudml.org/doc/272063

ER -

## References

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- [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. Zbl0359.60018MR448705
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- [8] Z. Šidák. Rectangular confidence regions for the means of multivariate normal distributions. J. Amer. Statist. Assoc.62 (1967) 626–633. Zbl0158.17705MR216666

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