On the Heyde theorem for discrete Abelian groups

Tom 177 / 2006

G. M. Feldman Studia Mathematica 177 (2006), 67-79 MSC: Primary 62E10; Secondary 60B15, 39B52. DOI: 10.4064/sm177-1-5


Let $X$ be a countable discrete Abelian group, ${\rm Aut} (X)$ the set of automorphisms of $X$, and $ I(X)$ the set of idempotent distributions on $X$. Assume that $\alpha_1, \alpha_2, \beta_1, \beta_2 \in {\rm Aut} (X)$ satisfy $\beta_1\alpha_1^{-1} \pm \beta_2\alpha_2^{-1} \in {\rm Aut} (X)$. Let $\xi_1, \xi_2$ be independent random variables with values in $X$ and distributions $\mu_1, \mu_2.$ We prove that the symmetry of the conditional distribution of $L_2 = \beta_1\xi_1 + \beta_2\xi_2$ given $L_1 = \alpha_1\xi_1 + \alpha_2\xi_2$ implies that $\mu_1, \mu_2 \in I(X)$ if and only if the group $X$ contains no elements of order two. This theorem can be considered as an analogue for discrete Abelian groups of the well-known Heyde theorem where the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form given another.


  • G. M. FeldmanMathematical Division
    B. Verkin Institute for Low Temperature Physics and Engineering
    National Academy of Sciences of Ukraine
    47, Lenin Ave., Kharkov, 61103, Ukraine

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