Sharp concentration phenomena in high-dimensional Orlicz balls

TL;DR

This paper investigates concentration phenomena and limit theorems for high-dimensional Orlicz balls, providing precise deviation formulas, a central limit theorem, and a strong law of large numbers for the distribution of vectors within these geometric objects.

Contribution

It introduces new deviation formulas, limit theorems, and concentration results specifically for high-dimensional Orlicz balls, advancing understanding of their probabilistic geometry.

Findings

Derived a precise deviation formula for intersections of Orlicz balls.

Established a central limit theorem in the critical case for the $W$-norm.

Proved a strong law of large numbers for the $W$-norm in high dimensions.

Abstract

In this article, we present a precise deviation formula for the intersection of two Orlicz balls generated by Orlicz functions $V$ and $W$. Additionally, we establish a (quantitative) central limit theorem in the critical case and a strong law of large numbers for the "$W$-norm" of the uniform distribution on $\mathbb{B}^{(n,V)}$. Our techniques also enable us to derive a precise formula for the thin-shell concentration of uniformly distributed random vectors in high-dimensional Orlicz balls. In our approach we establish an Edgeworth-expansion using methods from harmonic analysis together with an exponential change of measure argument.