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

Hueter, Irene

doi:10.1090/S0002-9947-99-02499-X

Limit theorems for the convex hull of random points in higher dimensions
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Trans. Amer. Math. Soc. 351 (1999), 4337-4363 Request permission

Abstract:

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

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