Limit theorems for the distance of random points in $l_p^n$-balls

TL;DR

This paper establishes a central limit theorem for the Euclidean distance between two independent random points in high-dimensional $l_p^n$-balls, extending to large deviations for certain $p$ values.

Contribution

It provides a new proof of the CLT for distances in $l_p^n$-balls and extends results to large deviation principles for $p \, \geq \, 2$.

Findings

CLT for distances in $l_p^n$-balls as dimension grows

Compact proof for the sphere case

Large deviation principles for $p \, \geq \, 2$

Abstract

In this paper, we prove that the Euclidean distance between two independent random vectors uniformly distributed on $l_p^n$-balls $(1 \leq p \leq \infty)$ or on its boundary satisfies a central limit theorem as $n$ tends to $\infty$. Also, we give a compact proof of the case of the sphere, which was proved by Hammersley. Furthermore, we complement our central limit theorem by providing large deviation principles for the cases $p \geq 2$.