The square negative correlation property for generalized Orlicz balls

TL;DR

This paper extends the negative correlation property of coordinate functions from $\, ext{ell}_p^n$ to generalized Orlicz balls, deriving concentration results and a CLT, with a counterexample for 1-symmetric bodies.

Contribution

It provides a simple, elementary proof of Euclidean concentration and CLT for generalized Orlicz balls, expanding understanding beyond prior complex proofs.

Findings

Negative correlation property extends to generalized Orlicz balls.

Euclidean norm exhibits concentration in these balls.

Counterexample shows the property does not hold for all 1-symmetric bodies.

Abstract

Antilla, Ball and Perissinaki proved that the squares of coordinate functions in $\ell_p^n$ are negatively correlated. This paper extends their results to balls in generalized Orlicz norms on R^n. From this, the concentration of the Euclidean norm and a form of the Central Limit Theorem for the generalized Orlicz balls is deduced. Also, a counterexample for the square negative correlation hypothesis for 1-symmetric bodies is given. Currently the CLT is known in full generality for convex bodies (see the paper "Power-law estimates for the central limit theorem for convex sets" by B. Klartag), while for generalized Orlicz balls a much more general result is true (see "The negative association property for the absolute values of random variables equidistributed on a generalized Orlicz ball" by M. Pilipczuk and J. O. Wojtaszczyk). While, however, both aforementioned papers are rather long, complicated and technical, this paper gives a simple and elementary proof of, eg., the Euclidean concentration for generalized Orlicz balls.