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THE SKITOVICH–DARMOIS THEOREM FOR LOCALLY COMPACT ABELIAN GROUPS

Part of: Abstract harmonic analysis Probability theory on algebraic and topological structures

Published online by Cambridge University Press:  11 June 2010

GENNADIY FELDMAN  and
PIOTR GRACZYK
GENNADIY FELDMAN*
Affiliation:
Mathematical Division, B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, 47 Lenin Ave, Kharkov 61103, Ukraine LAREMA Laboratoire de Mathématiques, Université d’Angers, 2 bd. Lavoisier, 49045 Angers, France (email: feldman@ilt.kharkov.ua)
PIOTR GRACZYK
Affiliation:
LAREMA Laboratoire de Mathématiques, Université d’Angers, 2 bd. Lavoisier, 49045 Angers, France (email: piotr.graczyk@univ-angers.fr)
*
✉For correspondence; e-mail: feldman@ilt.kharkov.ua
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Abstract

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According to the Skitovich–Darmois theorem, the independence of two linear forms of n independent random variables implies that the random variables are Gaussian. We consider the case where independent random variables take values in a second countable locally compact abelian group X, and coefficients of the forms are topological automorphisms of X. We describe a wide class of groups X for which a group-theoretic analogue of the Skitovich–Darmois theorem holds true when n=2.

Keywords

locally compact abelian groupSkitovich–Darmois theorem

MSC classification

Secondary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 43A25: Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 88 , Issue 3 , June 2010 , pp. 339 - 352
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

Footnotes

This research was supported in part by the Ukrainian–French grant PICS No 4769 (2009–2011).

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