A Simpler Proof of the Negative Association Property for Absolute Values of Measures Tied to Generalized Orlicz Balls

Volume 57 / 2009

Jakub Onufry Wojtaszczyk Bulletin Polish Acad. Sci. Math. 57 (2009), 41-56 MSC: Primary 52A20; Secondary 60D05. DOI: 10.4064/ba57-1-5

Abstract

Negative association for a family of random variables $(X_i)$ means that for any coordinatewise increasing functions $f,g$ we have $$\mathbb{E} f(X_{i_1},\ldots,X_{i_k}) g(X_{j_1},\ldots,X_{j_l}) \leq \mathbb{E} f(X_{i_1},\ldots,X_{i_k}) \mathbb{E} g(X_{j_1},\ldots,X_{j_l})$$ for any disjoint sets of indices $(i_m)$, $(j_n)$. 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.

Authors

  • Jakub Onufry WojtaszczykDepartment of Mathematics,
    Computer Science and Mechanics
    University of Warsaw
    Banacha 2
    02-097 Warszawa, Poland
    e-mail

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