Asymptotic distribution of the maximum interpoint distance in a sample of random vectors with a spherically symmetric distribution

TL;DR

This paper investigates the asymptotic behavior of the maximum interpoint distance in large samples of spherically symmetric random vectors, establishing Gumbel-type limit laws and extending existing results in multivariate extreme value theory.

Contribution

It generalizes previous findings by deriving Gumbel-type limit distributions for maximum interpoint distances in multidimensional spherically symmetric distributions.

Findings

Maximum interpoint distance follows a Gumbel-type limit law.

Results extend univariate extreme value theory to multivariate spherically symmetric cases.

Discussion of other limit laws and open problems included.

Abstract

Extreme value theory is part and parcel of any study of order statistics in one dimension. Our aim here is to consider such large sample theory for the maximum distance to the origin, and the related maximum "interpoint distance," in multidimensions. We show that for a family of spherically symmetric distributions, these statistics have a Gumbel-type limit, generalizing several existing results. We also discuss the other two types of limit laws and suggest some open problems. This work complements our earlier study on the minimum interpoint distance.