A conditional limit theorem for high-dimensional $\ell^{p}$ spheres

TL;DR

This paper proves a limit theorem for distributions on high-dimensional $\, ext{l}^p$ spheres conditioned on rare events, revealing their asymptotic behavior and tail properties in high-dimensional geometry.

Contribution

It introduces a new limit theorem for conditioned distributions on $\, ext{l}^p$ spheres in high dimensions, along with a large deviation principle for tail analysis.

Findings

Established a limit theorem for conditioned distributions on $\, ext{l}^p$ spheres.

Derived a large deviation principle relevant to tail behavior of random projections.

Provided insights into the asymptotic geometry of high-dimensional $\, ext{l}^p$ spaces.

Abstract

The study of high-dimensional distributions is of interest in probability theory, statistics and asymptotic convex geometry, where the object of interest is the uniform distribution on a convex set in high dimensions. The $\ell^p$ spaces and norms are of particular interest in this setting. In this paper, we establish a limit theorem for distributions on $\ell^p$ spheres, conditioned on a rare event, in a high-dimensional geometric setting. As part of our proof, we establish a certain large deviation principle that is also relevant to the study of the tail behavior of random projections of $\ell^p$ balls in a high-dimensional Euclidean space.