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Convex Hulls of Uniform Samples from a Convex Polygon

Published online by Cambridge University Press:  04 January 2016

Piet Groeneboom
Piet Groeneboom*
Affiliation:
Delft University of Technology
*
∗ Postal address: DIAM, Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands. Email address: p.groeneboom@tudelft.
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Abstract

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In Groeneboom (1988) a central limit theorem for the number of vertices Nn of the convex hull of a uniform sample from the interior of a convex polygon was derived. To be more precise, it was shown that {Nn - (2/3)rlogn} / {(10/27)rlogn}1/2 converges in law to a standard normal distribution, if r is the number of vertices of the convex polygon from which the sample is taken. In the unpublished preprint Nagaev and Khamdamov (1991) a central limit result for the joint distribution of Nn and An is given, where An is the area of the convex hull, using a coupling of the sample process near the border of the polygon with a Poisson point process as in Groeneboom (1988), and representing the remaining area in the Poisson approximation as a union of a doubly infinite sequence of independent standard exponential random variables. We derive this representation from the representation in Groeneboom (1988) and also prove the central limit result of Nagaev and Khamdamov (1991), using this representation. The relation between the variances of the asymptotic normal distributions of the number of vertices and the area, established in Nagaev and Khamdamov (1991), corresponds to a relation between the actual sample variances of Nn and An in Buchta (2005). We show how these asymptotic results all follow from one simple guiding principle. This corrects at the same time the scaling constants in Cabo and Groeneboom (1994) and Nagaev (1995).

Keywords

Convex hull
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 44 , Issue 2 , June 2012 , pp. 330 - 342
Copyright
© Applied Probability Trust 

Footnotes

Dedicated to the memories of Alexander Nagaev and Tomasz Schreiber.

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