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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Author: Irene Hueter
Journal: Trans. Amer. Math. Soc. 351 (1999), 4337-4363
MSC (1991): Primary 52A22, 60D05
Published electronically: July 21, 1999
MathSciNet review: 1670156
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Abstract | References | Similar Articles | Additional Information

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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Additional Information

Irene Hueter
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611
Email: hueter@math.ufl.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02499-X
PII: S 0002-9947(99)02499-X
Keywords: Convex hull, Poisson point process, Markovian jump process, (sub/super)-martingales, central limit theorem, rotationally invariant distributions.
Received by editor(s): December 2, 1998
Received by editor(s) in revised form: January 22, 1999
Published electronically: July 21, 1999
Article copyright: © Copyright 1999 American Mathematical Society



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