On Heyde’s Theorem for Probability Distributions on a Discrete Abelian Group

Let X be a countable discrete Abelian group containing no elements of order 2. Let α be an automorphism of X . Let ξ 1 and ξ 2 be independent random variables with values in the group X and distributions μ 1 and μ 2 . The main result of the article is the following statement. The symmetry of the con...

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Bibliographic Details
Published in:Doklady. Mathematics 2018, Vol.97 (1), p.11-14
Main Author: Feldman, G. M.
Format: Article
Language:English
Subjects:
Automorphisms
Independent variables
Mathematics
Mathematics and Statistics
Random variables
Subgroups
Theorems
Citations: Items that this one cites
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Summary:Let X be a countable discrete Abelian group containing no elements of order 2. Let α be an automorphism of X . Let ξ 1 and ξ 2 be independent random variables with values in the group X and distributions μ 1 and μ 2 . The main result of the article is the following statement. The symmetry of the conditional distribution of the linear form L 2 = ξ 1 + αξ 2 given L 1 = ξ 1 + ξ 2 implies that μ j are shifts of the Haar distribution of a finite subgroup of X if and only if α satisfies the condition Ker( I + α)= {0}. Some generalisations of this theorem are also proved.
ISSN:1064-5624
1531-8362
DOI:10.1134/S1064562418010027