Limit laws for the diameter of a set of random points from a distribution supported by a smoothly bounded set

TL;DR

This paper derives limit laws for the maximum distance between random points in a smooth, bounded set, extending classical results to Poisson processes and specific distributions like the Pearson type II.

Contribution

It introduces new asymptotic results for the maximum interpoint distance in Poisson point processes within smooth bounded sets, including ellipsoids and Pearson type II distributions.

Findings

Limit law for points in a $d$-dimensional ellipsoid.

Extension to Pearson type II distribution.

Asymptotic behavior characterized for smooth boundary sets.

Abstract

We study the asymptotic behavior of the maximum interpoint distance of random points in a $d$-dimensional set with a unique diameter and a smooth boundary at the poles. Instead of investigating only a fixed number of $n$ points as $n$ tends to infinity, we consider the much more general setting in which the random points are the supports of appropriately defined Poisson processes. The main result covers the case of uniformly distributed points within a $d$-dimensional ellipsoid with a unique major axis. Moreover, several generalizations of the main result are established, for example a limit law for the maximum interpoint distance of random points from a Pearson type II distribution.