Limit theorems for the distance between two 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 their boundaries according to the cone measure satisfies a central limit theorem as $n\to \infty$. Also, we give a compact proof of the case of the sphere, which was proved by Hammersley. Furthermore, we complement our central limit theorems by providing large deviation principles for points in the balls and on their boundaries when $1\le p\le \infty$. The speed is $n$ for $p\ge 2$, whereas the regime $1\le p<2$ has speed $n^{p/2}$ and an explicit rate function.