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 Teor. Veroyatnost. i Primenen., 2018, Volume 63, Issue 4, Pages 730–754 (Mi tvp5163)

On a characterization theorem for probability distributions on discrete Abelian groups

G. M. Feldman

B. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, Khar'kov

Abstract: Let $X$ be a countable discrete Abelian group containing no elements of order 2, $\alpha$ be an automorphism of $X$, and $\xi_1$ and $\xi_2$ be independent random variables with values in the group $X$ and having distributions $\mu_1$ and $\mu_2$. The main result of the present paper is as follows. The symmetry of the conditional distribution of the linear form $L_2 = \xi_1 + \alpha\xi_2$ given $L_1 = \xi_1 + \xi_2$ implies that $\mu_j$ are shifts of the Haar distribution of a finite subgroup of $X$ if and only if the automorphism $\alpha$ satisfies the condition $\operatorname{Ker}(I+\alpha)=\{0\}$. This theorem is an analogue, for discrete Abelian groups, of the well-known Heyde theorem, where a Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form of independent random variable given the other. We also prove some generalizations of this theorem.

Keywords: conditional distribution, Haar distribution, discrete Abelian group.

DOI: https://doi.org/10.4213/tvp5163

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English version:
Theory of Probability and its Applications, 2019, 63:4, 594–612

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Accepted:24.04.2018

Citation: G. M. Feldman, “On a characterization theorem for probability distributions on discrete Abelian groups”, Teor. Veroyatnost. i Primenen., 63:4 (2018), 730–754; Theory Probab. Appl., 63:4 (2019), 594–612

Citation in format AMSBIB
\Bibitem{Fel18} \by G.~M.~Feldman \paper On a characterization theorem for probability distributions on discrete Abelian groups \jour Teor. Veroyatnost. i Primenen. \yr 2018 \vol 63 \issue 4 \pages 730--754 \mathnet{http://mi.mathnet.ru/tvp5163} \crossref{https://doi.org/10.4213/tvp5163} \elib{https://elibrary.ru/item.asp?id=36361411} \transl \jour Theory Probab. Appl. \yr 2019 \vol 63 \issue 4 \pages 594--612 \crossref{https://doi.org/10.1137/S0040585X97T989271} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000457756800006} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85064592802} 

• http://mi.mathnet.ru/eng/tvp5163
• https://doi.org/10.4213/tvp5163
• http://mi.mathnet.ru/eng/tvp/v63/i4/p730

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This publication is cited in the following articles:
1. Feldman G.M., “On a Characterization Theorem For Locally Compact Abelian Groups Containing An Element of Order 2”, Potential Anal.
2. G. Feldman, “On a characterization theorem for connected locally compact abelian groups”, J. Fourier Anal. Appl., 26:1 (2020), 14
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