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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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References

1. Artstein, V. and Vitale, R. A. (1975) A strong law of large numbers for random compact sets. Ann. Prob. 3, 879–882.Google Scholar
2. Bonnesen, T. and Fenchel, W. (1934) Theorie der Konvexen Körper. Springer-Verlag, Berlin.Google Scholar
3. Bourbaki, N. (1969) Éléments de Mathématique, Intégration, Chapitre 9, Mesures sur les espaces topologiques séparés, première édition. Hermann, Paris.Google Scholar
4. Christensen, J. P. R. (1974) Topology and Borel Structure. North-Holland, Amsterdam.Google Scholar
5. Hadwiger, H. (1957) Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. Springer-Verlag, Berlin.Google Scholar
6. Jordan, C. (1950) Calculus of Finite Differences. Chelsea, New York.Google Scholar
7. Kendall, D. G. (1974) Foundations of a theory of random sets. In Stochastic Geometry, ed. Harding, E. H. and Kendall, D. G., Wiley, London, pp. 322–376.Google Scholar
8. Lukacs, E. (1969) Characteristic Functions, second edition. Griffin, London.Google Scholar
9. Matheron, G. (1975) Random Sets and Integral Geometry. Wiley, London.Google Scholar
10. Parthasarathy, K. R. (1967) Probability Measures on Metric Spaces. Academic Press, New York.Google Scholar
11. Radström, H. (1952) An embedding theorem for spaces of convex sets. Proc. Amer. Math. Soc. 3, 165–169.CrossRefGoogle Scholar
12. Santaló, L. A. (1976) Integral Geometry and Geometrical Probability. Addison-Wesley, London.Google Scholar
13. Streit, F. (1970) On multiple integral geometric integrals and their applications to probability theory. Canad. J. Math. 22, 151–163.Google Scholar