Limit theorems for the convex hull of random points in higher dimensions

Abstract

We give a central limit theorem for the number N n N_n of vertices of the convex hull of n n independent and identically distributed random vectors, being sampled from a certain class of spherically symmetric distributions in R d ( d > 1 ) , \mathbb {R}^d \; (d> 1), that includes the normal family. Furthermore, we prove that, among these distributions, the variance of N n N_n exhibits the same order of magnitude as the expectation as n → ∞ . n \rightarrow \infty . The main tools are Poisson approximation of the point process of vertices of the convex hull and (sub/super)-martingales.