On the Heyde theorem for discrete Abelian groups

G. M. Feldman¹

¹ Mathematical Division B. Verkin Institute for Low Temperature Physics and Engineering National Academy of Sciences of Ukraine 47, Lenin Ave., Kharkov, 61103, Ukraine

Studia Mathematica, Tome 177 (2006) no. 1, pp. 67-79

Résumé

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.

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DOI : 10.4064/sm177-1-5

Keywords: countable discrete abelian group aut set automorphisms set idempotent distributions assume alpha alpha beta beta aut satisfy beta alpha beta alpha aut independent random variables values distributions prove symmetry conditional distribution beta beta given alpha alpha implies only group contains elements order theorem considered analogue discrete abelian groups well known heyde theorem where gaussian distribution real line characterized symmetry conditional distribution linear form given another