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

Jakub Onufry Wojtaszczyk¹

¹ Department of Mathematics, Computer Science and Mechanics University of Warsaw Banacha 2 02-097 Warszawa, Poland

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 57 (2009) no. 1, pp. 41-56

Résumé

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.

Détail

DOI : 10.4064/ba57-1-5

Keywords: negative association family random variables means coordinatewise increasing functions have mathbb ldots ldots leq mathbb ldots mathbb ldots disjoint sets indices indicate negative correlation family random variables first introduced statistics alem amp saxena joag dev amp proschan brought convex geometry wojtaszczyk amp pilipczuk prove central limit theorem orlicz balls paper gives relatively simple proof negative association absolute values wide class measures tied generalized orlicz balls including uniform measures balls