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

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 Poincaré-Maxwell-Borel lemma, we deduce a conditional limit theorem for highdimensional 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 Poincaré-Maxwell-Borel lemma, and thus the geometric interpretation, undergoes a phase transition depending on the magnitude of the V 2 -Orlicz radius.

A probabilistic approach to Lorentz balls

“…Recently, Kabluchko and Prochno [31], using maximum entropy considerations in the framework of non-interacting particles that have their origin in statistical mechanics, derived an asymptotic version of a Schechtman-Zinn type representation for Orlicz spaces, relating the uniform distribution on Orlicz balls to Gibbs distributions with potential given by the Orlicz functions. This connection allowed to obtain a number of further results in the general setting of Orlicz balls, wich can be found, for instance, in [2,8,21,28,35].…”

“…To apply variants of the Gibbs conditioning principle as the one used, e.g., in [21] and [36], we have to account for the fact that the set K n depends on n. For large n it will be approximately equal to the set…”

“…The motivation of our paper is thus twofold, namely to develop a first probabilistic approach on the one hand and apply this to study some asymptotic properties of Lorentz spaces n q,p on the other. In view of the vast literature regarding a probabilistic take on n p or more generally n M spaces and their unit balls, this is a first step towards approaching numerous natural questions that arise, like central limit theorems, conditional limit theorems, or moderate and large deviations [3,4,21,22,30,32,36], asymptotic volume ratios and thin-shell measure concentration [2,31], asymptotic independence of coordinates [8] or asymptotic thin-shell results [35], just to mention a few.…”