Limit theorems for random simplices in high dimensions

Abstract

Let $r=r(n)$ be a sequence of integers such that $r\le n$ and let $X_1,\dots ,X_{r+1}$ 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 $X_1,\dots ,X_{r+1}$ 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-$\varphi$ 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 $0<\alpha <1$).