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

Abstract

In this paper, we prove that the Euclidean distance between two independent random vectors uniformly distributed on $l_p^n$-balls $(1\le p\le \infty )$ 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 $p\ge 2$.