The volume of random simplices from elliptical distributions in high dimension

TL;DR

This paper investigates the asymptotic behavior of the volume of high-dimensional random simplices generated by elliptical distributions, establishing limit theorems for their logarithmic volume and related log-determinants.

Contribution

It introduces new limit theorems for the volume of random simplices from elliptical distributions in high dimensions, extending understanding of their probabilistic properties.

Findings

Central limit theorem for the log-volume of random simplices

Stable limit theorem in high-dimensional regime

Limit theorem for the log-determinant of elliptical random matrices

Abstract

Random simplices and more general random convex bodies of dimension $p$ in $\mathbb{R}^n$ with $p\leq n$ are considered, which are generated by random vectors having an elliptical distribution. In the high-dimensional regime, that is, if $p\to\infty$ and $n\to\infty$ in such a way that $p/n\to\gamma\in(0,1)$, a central and a stable limit theorem for the logarithmic volume of random simplices and random convex bodies is shown. The result follows from a related central limit theorem for the log-determinant of $p\times n$ random matrices whose rows are copies of a random vector with an elliptical distribution, which is established as well.