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Limit theorems for convex hulls of random sets

Part of: Stochastic processes Geometric probability and stochastic geometry Real functions: Miscellaneous topics

Published online by Cambridge University Press:  01 July 2016

Ilya S. Molchanov
Ilya S. Molchanov*
Affiliation:
Kiev Technological Institute
*
∗Permanent address: Kiev Technological Institute of Food Industry, Vladimirskaya, 68, Kiev, 252017, Ukraine. Currently visiting Bergakademie Freiberg, Germany.
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Abstract

Let , be i.i.d. random closed sets in . Limit theorems for their normalized convex hulls conv () are proved. The limiting distributions correspond to C-stable random sets. The random closed set A is called C-stable if, for any , the sets anA and conv ( coincide in distribution for certain positive an, compact Kn, and independent copies A1, …, An of A. The distributions of C-stable sets are characterized via corresponding containment functionals.

Keywords

CONTAINMENT FUNCTIONALSTABLE DISTRIBUTIONREGULAR VARIATIONRANDOM SETPROBABILITY DISTRIBUTIONMAX-STABILITYRANDOM SAMPLEMULTIVALUED FUNCTION

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 26E25: Set-valued functions 60G70: Extreme value theory; extremal processes
Type
Research Article
Information
Advances in Applied Probability , Volume 25 , Issue 2 , June 1993 , pp. 395 - 414
Copyright
Copyright © Applied Probability Trust 1993 

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