Limit theorems for random simplices in high dimensions

TL;DR

This paper establishes limit theorems for the volume and log-volume of random simplices in high-dimensional spaces, considering various distributions and growth regimes of the number of points relative to dimension.

Contribution

It provides new limit theorems including CLTs, log-normal, and deviation results for random simplices in high dimensions, depending on the growth rate of r relative to n.

Findings

Normal fluctuations when r=o(n)

Log-normal fluctuations when r~αn for 0<α<1

Different types of mod-φ convergence are derived

Abstract

Let $r=r(n)$ be a sequence of integers such that $r\leq n$ and let $X_1,\ldots,X_{r+1}$ be independent random points distributed according to the Gaussian, the Beta or the spherical distribution on $\mathbb{R}^n$. Limit theorems for the log-volume and the volume of the random convex hull of $X_1,\ldots,X_{r+1}$ are established in high dimensions, that is, as $r$ and $n$ tend to infinity simultaneously. This includes, Berry-Esseen-type central limit theorems, log-normal limit theorems, moderate and large deviations. Also different types of mod-$\phi$ convergence are derived. The results heavily depend on the asymptotic growth of $r$ relative to $n$. For example, we prove that the fluctuations of the volume of the simplex are normal (respectively, log-normal) if $r=o(n)$ (respectively, $r\sim \alpha n$ for some $0 < \alpha < 1$).