Sanov-type large deviations and conditional limit theorems for high-dimensional Orlicz balls

TL;DR

This paper establishes a large deviation principle and a conditional limit theorem for high-dimensional vectors in Orlicz balls, revealing phase transitions in their geometric structure based on Orlicz radii.

Contribution

It introduces a Sanov-type large deviation principle for empirical measures in Orlicz balls and derives a conditional limit theorem showing phase transitions in high-dimensional geometry.

Findings

Large deviation principle for empirical measures in Orlicz balls

Conditional limit theorem with phase transition behavior

Geometric interpretation of high-dimensional Orlicz ball distributions

Abstract

In this paper, we prove a Sanov-type large deviation principle for the sequence of empirical measures of vectors chosen uniformly at random from an Orlicz ball. From this level-$2$ large deviation result, in a combination with Gibbs conditioning, entropy maximization and an Orlicz version of the Poincar\'e-Maxwell-Borel lemma, we deduce a conditional limit theorem for high-dimensional Orlicz balls. Roughly speaking, the latter shows that if $V_1$ and $V_2$ are Orlicz functions, then random points in the $V_1$-Orlicz ball, conditioned on having a small $V_2$-Orlicz radius, look like an appropriately scaled $V_2$-Orlicz ball. In fact, we show that the limiting distribution in our Poincar\'e-Maxwell-Borel lemma, and thus the geometric interpretation, undergoes a phase transition depending on the magnitude of the $V_2$-Orlicz radius.