Limit theorems for functionals of convex hulls

A. J. Cabo*, P. Groeneboom

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

48 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)31-55
Number of pages25
JournalProbability Theory and Related Fields
Issue number1
Publication statusPublished - 1994
Externally publishedYes


  • Mathematics Subject Classifications (1991): 52A22, 52A38, 60G44, 60G55


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