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HEYDE’S CHARACTERIZATION THEOREM FOR DISCRETE ABELIAN GROUPS

Part of: Probability theory on algebraic and topological structures

Published online by Cambridge University Press:  22 January 2010

MARGARYTA MYRONYUK
MARGARYTA MYRONYUK*
Affiliation:
Mathematical Division, B. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, 47 Lenin Avenue, Kharkov 61103, Ukraine (email: myronyuk@ilt.kharkov.ua)
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Abstract

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Let X be a countable discrete abelian group with automorphism group Aut(X). Let ξ1 and ξ2 be independent X-valued random variables with distributions μ1 and μ2, respectively. Suppose that α1,α2,β1,β2∈Aut(X) and β1α−11±β2α−12∈Aut(X). Assuming that the conditional distribution of the linear form L2 given L1 is symmetric, where L2=β1ξ1+β2ξ2 and L1=α1ξ1+α2ξ2, we describe all possibilities for the μj. This is a group-theoretic analogue of Heyde’s characterization of Gaussian distributions on the real line.

Keywords

characterization theoremdiscrete abelian groupHeyde’s theorem

MSC classification

Secondary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 88 , Issue 1 , February 2010 , pp. 93 - 102
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

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