The diameter of a random elliptical cloud

TL;DR

This paper investigates the asymptotic behavior of the maximum interpoint distance in large clouds of i.i.d. elliptical vectors, revealing non-standard extreme value phenomena and the role of norm attraction properties.

Contribution

It provides comprehensive results on the diameter distribution of elliptical vector clouds, especially when norms are in the Gumbel domain of attraction, including generalizations to other norms and dimensions.

Findings

Diameter follows non-standard extreme value limits.

Localization into low-dimensional subspaces occurs for large norms.

Results extend to various norms and bi-dimensional cases.

Abstract

We study the asymptotic behavior of the diameter or maximum interpoint distance of a cloud of i.i.d. $d$-dimensional random vectors when the number of points in the cloud tends to infinity. This is a non standard extreme value problem since the diameter is a max-$U$-statistic, hence a maximum of dependent random variables. Therefore, the limiting distributions may not be extreme value distributions. We obtain exhaustive results for the Euclidean diameter of a cloud of elliptical vectors whose Euclidean norm is in the domain of attraction for the maximum of the Gumbel distribution. We also obtain results in other norms for spherical vectors and we give several bi-dimensional generalizations. The main idea behind our results and their proofs is a specific property of random vectors whose norm is in the domain of attraction of the Gumbel distribution: the localization into subspaces of low dimension of vectors with a large norm.