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Factorization of probability measures on symmetric hypergroups

Part of: Distribution theory - Probability Probability theory on algebraic and topological structures

Published online by Cambridge University Press:  09 April 2009

Michael Voit
Michael Voit
Affiliation:
Mathematisches Institut Technische Universität MünchenArcisstr. 21 D-8000 München 2, Germany
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Abstract

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Generalizing known results for special examples, we derive a Khintchine type decomposition of probability measures on symmetric hypergroups. This result is based on a triangular central limit theorem and a discussion of conditions ensuring that the set of all factors of a probability measure is weakly compact. By our main result, a probability measure satisfying certain restrictions can be written as a product of indecomposable factors and a factor in I0(K), the set of all measures having decomposable factors only. Some contributions to the classification of I0(K) are given for general symmetric hypergroups and applied to several families of examples like finite symmetric hypergroups and hypergroup joins. Furthermore, all results are discussed in detail for a class of discrete symmetric hypergroups which are generated by infinitely many joins, for a class of countable compact hypergroups, for Sturm-Liouville hypergroups on [0, ∞[ and, finally, for polynomial hypergroups.

Keywords

Khintchine decompositioninfinitely divisible measuresindecomposable measuressymmetric hypergroupspolynomial hypergroupsSturm-Liouville hypergroupshypergroup joins

MSC classification

Secondary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 60E07: Infinitely divisible distributions; stable distributions 60E10: Characteristic functions; other transforms
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 50 , Issue 3 , June 1991 , pp. 417 - 467
Copyright
Copyright © Australian Mathematical Society 1991

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