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.
Article PDF
Similar content being viewed by others
References
Buchta, C.: Stochastische Approximation konvexer Polygone. Z. Wahrscheinlichkeitstheor. Verw. Geb.67, 283–304 (1984)
Efron, B.: The convex hull of a random set of points Biometrika52, 331–343 (1965)
Feller, W.: Introduction to Probability Theory and its Applications (2nd ed.) New York: Wiley 1971
Groeneboom, P.: Limit Theorems for Convex Hulls. Probab. Theory Relat. Fields79, 327–368 (1988)
Hueter, I.: The convex hull ofn random points and its vertex process. Doctoral Dissertation. University of Berne, 1992
Ibragimov, I.A., Linnik, Y.U.: Independent and stationary sequences of random variables. Groningen: Wolters Noordhoff 1971
Rényi, A., Sulanke, R.: Über die konvexe Hülle vonn zufällig gewählten Punkten. Z. Wahrscheinlichkeitstheor. Verw. Geb.2, 75–84 (1963)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Cabo, A.J., Groeneboom, P. Limit theorems for functionals of convex hulls. Probab. Th. Rel. Fields 100, 31–55 (1994). https://doi.org/10.1007/BF01204952
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01204952