A simpler proof of the negative association property for absolute values of measures tied to generalized Orlicz balls

• Autorzy: Wojtaszczyk J. O.
• Języki publikacji: EN

Abstrakty

EN

Negative association for a family of random variables (Xi) means that for any coordinatewise increasing functions ƒ, g we have Eƒ(Xi1,...,Xik)g(Xj1,...,Xjt) ≤ E ƒ(Xi1,...,Xik)Eg(Xj1,...,Xjt) for any disjoint sets of indices (im ), (jn ). It is a way to indicate the negative correlation in a family of random variables. It was first introduced in 1980s in statistics by Alem & Saxena and Joag-Dev & Proschan, and brought to convex geometry in 2005 by Wojtaszczyk & Pilipczuk to prove the Central Limit Theorem for Orlicz balls. The paper gives a relatively simple proof of negative association of absolute values for a wide class of measures tied to generalized Orlicz balls, including the uniform measures on such balls.

Słowa kluczowe

EN

negative association; Orlicz balls

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2009
• Tom: Vol. 57, no 1
• Strony: 41--56
• Opis fizyczny: Bibliogr. 7 poz.

Twórcy

autor

Wojtaszczyk J. O.

• Department of Mathematics, Computer Science and Mechanics, University of Warsaw, Banacha 2 02-097 Warszawa, Poland,

Bibliografia

• [1] K. Alam and K. M. L. Saxena, Positive dependence in multivariate distributions, Comm. Statist. Theory Methods A 10 (1981), 1183-1196.
• [2] B. Fleury, Between Paouris concentration inequality and variance conjecture, preprint.
• [3] R. J. Gardner, The Brunn-Minkowski inequality, Bull. Amer. Math. Soc. 39 (2002), 355-405.
• [4] K. Joag-Dev and F. Proschan, Negative association of random variables with applications, Ann. Statist. 11 (1983), 286-295.
• [5] R. Kannan, L. Lovasz and M. Simonovits, Isoperimetric problems for convex bodies and a localization lemma, Discrete Comput. Geom. 13 (1995), 541-559.
• [6] M. Pilipczuk and J. O. Wojtaszczyk, The negative association property for the absolute values of random variables equidistributed on a generalized Orlicz ball, Positivity 12 (2008), 421-474.
• [7] J. O. Wojtaszczyk, The square negative correlation property for generalized Orlicz balls, in: Geometric Aspects of Functional Analysis (Israel Seminar, 2004-2005), Lecture Notes in Math. 1910, Springer, 2007, 305-313.