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Convergence in mean of some characteristics of the convex hull

Published online by Cambridge University Press:  01 July 2016

Henk Brozius
Henk Brozius*
Affiliation:
Erasmus University Rotterdam
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Abstract

A sequence Xn, 1 of independent and identically distributed random vectors is considered. Under a condition of regular variation, the number of vertices of the convex hull of {X1, …, Xn} converges in distribution to the number of vertices of the convex hull of a certain Poisson point process. In this paper, it is proved without sharpening the conditions that the expectation of this number also converges; expressions are found for its limit, generalizing results of Davis et al. (1987). We also present some results concerning other quantities of interest, such as area and perimeter of the convex hull and the probability that a given point belongs to the convex hull.

Keywords

REGULAR VARIATIONPOINT PROCESSESPALM PROBABILITIES
Type
Research Article
Information
Advances in Applied Probability , Volume 21 , Issue 3 , September 1989 , pp. 526 - 542
Copyright
Copyright © Applied Probability Trust 1989 

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Footnotes

This study was funded by the Netherlands Organization for the Advancement of Pure Research (N.W.O.).

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