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

• Volume: 57, Issue: 1, page 41-56
• ISSN: 0239-7269

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## Abstract

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Negative association for a family of random variables $\left({X}_{i}\right)$ means that for any coordinatewise increasing functions f,g we have $\left({X}_{i₁},...,{X}_{{i}_{k}}\right)g\left({X}_{j₁},...,{X}_{{j}_{l}}\right)\le f\left({X}_{i₁},...,{X}_{{i}_{k}}\right)g\left({X}_{j₁},...,{X}_{{j}_{l}}\right)$ 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.

## How to cite

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Jakub Onufry Wojtaszczyk. "A Simpler Proof of the Negative Association Property for Absolute Values of Measures Tied to Generalized Orlicz Balls." Bulletin of the Polish Academy of Sciences. Mathematics 57.1 (2009): 41-56. <http://eudml.org/doc/281206>.

@article{JakubOnufryWojtaszczyk2009,
abstract = {Negative association for a family of random variables $(X_i)$ means that for any coordinatewise increasing functions f,g we have $(X_\{i₁\},...,X_\{i_k\}) g(X_\{j₁\},...,X_\{j_l\}) ≤ f(X_\{i₁\},...,X_\{i_k\}) g(X_\{j₁\},...,X_\{j_l\})$ 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.},
author = {Jakub Onufry Wojtaszczyk},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
keywords = {negative association; Orlicz balls; central limit theorem; log-concave measure},
language = {eng},
number = {1},
pages = {41-56},
title = {A Simpler Proof of the Negative Association Property for Absolute Values of Measures Tied to Generalized Orlicz Balls},
url = {http://eudml.org/doc/281206},
volume = {57},
year = {2009},
}

TY - JOUR
AU - Jakub Onufry Wojtaszczyk
TI - A Simpler Proof of the Negative Association Property for Absolute Values of Measures Tied to Generalized Orlicz Balls
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2009
VL - 57
IS - 1
SP - 41
EP - 56
AB - Negative association for a family of random variables $(X_i)$ means that for any coordinatewise increasing functions f,g we have $(X_{i₁},...,X_{i_k}) g(X_{j₁},...,X_{j_l}) ≤ f(X_{i₁},...,X_{i_k}) g(X_{j₁},...,X_{j_l})$ 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.
LA - eng
KW - negative association; Orlicz balls; central limit theorem; log-concave measure
UR - http://eudml.org/doc/281206
ER -

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