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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

References

[1]Feldman, G. M., ‘Marcinkiewicz and Lukacs theorems on abelian groups’, Theory Probab. Appl. 34 (1989), 290–297.Google Scholar
[2]Feldman, G. M., Arithmetic of Probability Distributions and Characterization Problems on Abelian Groups, Translations of Mathematical Monographs, 116 (American Mathematical Society, Providence, RI, 1993).Google Scholar
[3]Feldman, G. M., ‘On the Heyde theorem for finite abelian groups’, J. Theoret. Probab. 17 (2004), 929–941.CrossRefGoogle Scholar
[4]Feldman, G. M., ‘On a characterization theorem for locally compact abelian groups’, Probab. Theory Related Fields 133 (2005), 345–357.CrossRefGoogle Scholar
[5]Feldman, G. M., ‘On the Heyde theorem for discrete abelian groups’, Studia Math. 177(1) (2006), 67–79.Google Scholar
[6]Feldman, G. M., Functional Equations and Characterization Problems on Locally Compact Abelian Groups, EMS Tracts in Mathematics, 5 (European Mathematical Society, Zürich, 2008).CrossRefGoogle Scholar
[7]Fuchs, L., Infinite Abelian Groups, Vol. 1 (Academic Press, New York, 1970).Google Scholar
[8]Fuchs, L., Infinite Abelian Groups, Vol. 2 (Academic Press, New York, 1973).Google Scholar
[9]Hewitt, E. and Ross, K. A., Abstract Harmonic Analysis, Vol. 1 (Springer, Berlin, 1963).Google Scholar
[10]Heyde, C. C., ‘Characterization of the normal law by the symmetry of a certain conditional distribution’, Sankhya Ser. A 32 (1970), 115–118.Google Scholar
[11]Kagan, A. M., Linnik, Yu. V. and Rao, C. R., Characterization Problems in Mathematical Statistics, Wiley Series in Probability and Mathematical Statistics (John Wiley & Sons, New York, 1973).Google Scholar
[12]Mironyuk, M. V. and Fel’dman, G. M., ‘On a characterization theorem on finite abelian groups’, Siberian Math. J. 46(2) (2005), 315–324.Google Scholar