Probability Theory and Related Fields

, Volume 100, Issue 1, pp 31–55

Limit theorems for functionals of convex hulls

  • A. J. Cabo
  • P. Groeneboom
Article

Summary

In [4] a central limit theorem for the number of vertices of the convex hull of a uniform sample from the interior of a convex polygon is derived. This is done by approximating the process of vertices of the convex hull by the process of extreme points of a Poisson point process and by considering the latter process of extreme points as a Markov process (for a particular parametrization). We show that this method can also be applied to derive limit theorems for the boundary length and for the area of the convex hull. This extents results of Rényi and Sulanke (1963) and Buchta (1984), and shows that the boundary length and the area have a strikingly different probabilistic behavior.

Mathematics Subject Classifications (1991)

52A22 52A38 60G44 60G55 

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Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • A. J. Cabo
    • 1
  • P. Groeneboom
    • 1
  1. 1.Mathematics DepartmentDelft University of TechnologyDelftThe Netherlands

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