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

TL;DR

This paper presents a simplified proof demonstrating the negative association property of absolute values for a broad class of measures related to generalized Orlicz balls, which has implications in convex geometry and probability theory.

Contribution

It provides a new, simpler proof of negative association for measures tied to generalized Orlicz balls, expanding understanding in convex geometry and probability.

Findings

Negative association holds for measures on generalized Orlicz balls

The proof applies to uniform measures on these balls

Simplifies previous complex proofs in the area

Abstract

Negative association for a family of random variables $(X_i)$ means that for any coordinate--wise increasing functions $f,g$ we have $$\E f(X_{i_1},...,X_{i_k}) g(X_{j_1},...,X_{j_l}) \leq \E f(X_{i_1},...,X_{i_k}) \E g(X_{j_1},...,X_{j_l})$$ 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, and brought to convex geometry in 2005 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 generalized Orlicz balls.