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

Abstract

top
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.

How to cite

top

Figalli, 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

top
  1. [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. Zbl1206.60011MR2649000
  2. [2] C. Borell. A Gaussian correlation inequality for certain bodies in 𝐑 n . Math. Ann. 256 (4) (1981) 569–573. Zbl0451.60018MR628236
  3. [3] G. Harge. A particular case of correlation inequality for the Gaussian measure. Ann. Probab. 27 (4) (1999) 1939–1951. Zbl0962.28013MR1742895
  4. [4] C. G. Khatri. On certain inequalities for normal distributions and their applications to simultaneous confidence bounds. Ann. Math. Statist.38 (1967) 1853–1867. Zbl0155.27103MR220392
  5. [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. [6] L. D. Pitt. A Gaussian correlation inequality for symmetric convex sets. Ann. Probab. 5 (3) (1977) 470–474. Zbl0359.60018MR448705
  7. [7] G. Schechtman, T. Schlumprecht and J. Zinn. On the Gaussian measure of the intersection. Ann. Probab. 26 (1) (1998) 346–357. Zbl0936.60015MR1617052
  8. [8] Z. Šidák. Rectangular confidence regions for the means of multivariate normal distributions. J. Amer. Statist. Assoc.62 (1967) 626–633. Zbl0158.17705MR216666

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.