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Random compact convex sets which are infinitely divisible with respect to Minkowski addition

Published online by Cambridge University Press:  01 July 2016

Shigeru Mase
Shigeru Mase*
Affiliation:
Tokyo Institute of Technology
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Abstract

Random closed sets (in Matheron's sense) which are a.s. compact convex and contain the origin are considered. The totality of such random closed sets are closed under the Minkowski addition and we can define the concept of infinite divisibility with respect to Minkowski addition of random compact convex sets. Using a generalized notion of Laplace transformations we get Lévy-type canonical representations of infinitely divisible random compact convex sets. Isotropic and stable cases are also considered. Finally we get several mean formulas of Minkowski functionals of infinitely divisible random compact convex sets in terms of their Lévy spectral measures.

Keywords

RANDOM CLOSED SETCONVEXITYMINKOWSKI ADDITIONINFINITE DIVISIBILITYLEVY CANONICAL REPRESENTATIONMINKOWSKI FUNCTIONALMEAN VALUE
Type
Research Article
Information
Advances in Applied Probability , Volume 11 , Issue 4 , December 1979 , pp. 834 - 850
Copyright
Copyright © Applied Probability Trust 1979 

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