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The convex hull of a normal sample

Part of: Geometric probability and stochastic geometry Markov processes Limit theorems General convexity Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Irene Hueter
Irene Hueter*
Affiliation:
Purdue University
*
* Present address: Department of Statistics, University of California, Berkeley, CA 94720, USA.
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Abstract

Consider the convex hull of n independent, identically distributed points in the plane. Functionals of interest are the number of vertices Nn, the perimeter Ln and the area An of the convex hull. We study the asymptotic behaviour of these three quantities when the points are standard normally distributed. In particular, we derive the variances of Nn, Ln and An for large n and prove a central limit theorem for each of these random variables. We enlarge on a method developed by Groeneboom (1988) for uniformly distributed points supported on a bounded planar region. The process of vertices of the convex hull is of central importance. Poisson approximation and martingale techniques are used.

Keywords

POISSON POINT PROCESSMARKOVIAN JUMP PROCESSMARTINGALESNORMALDISTRIBUTIONCENTRAL LIMIT THEOREMVARIANCE

MSC classification

Primary: 52A22: Random convex sets and integral geometry
Secondary: 60D05: Geometric probability and stochastic geometry 60J75: Jump processes 60F05: Central limit and other weak theorems 60G44: Martingales with continuous parameter 60G55: Point processes
Type
Research Article
Information
Advances in Applied Probability , Volume 26 , Issue 4 , December 1994 , pp. 855 - 875
Copyright
Copyright © Applied Probability Trust 1994 

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Footnotes

Research supported by Swiss NSF Grant 20-28927.90.

References

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