Advances in Applied Probability

Convex hulls of uniform samples from a convex polygon

Piet Groeneboom

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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 In the unpublished preprint Nagaev and Khamdamov (1991) a central limit result for the joint distribution of Nn and An is given, where An is the area of the convex hull, using a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).

Article information

Adv. in Appl. Probab., Volume 44, Number 2 (2012), 330-342.

First available in Project Euclid: 16 June 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60E20
Secondary: 49G03 49F10

Convex hull


Groeneboom, Piet. Convex hulls of uniform samples from a convex polygon. Adv. in Appl. Probab. 44 (2012), no. 2, 330--342. doi:10.1239/aap/1339878714.

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