Heyde's characterisation theorem for discrete Abelian Groups

Margaryta Myronyuk


Let $X$ be a countable discrete Abelian group, and ${\rm Aut}(X)$
be the automorphism group of $X$. Let $\xi_1,\xi_2$ be independent
random variables with values in $X$ and distributions
$\mu_1,\mu_2$. Suppose that $\alpha_1,\alpha_2,\beta_1,\beta_2 \in
{\rm Aut}(X)$ and $\beta_1\alpha^{-1}_1 \pm \beta_2\alpha^{-1}_2
\in {\rm Aut}(X)$. Assume that the conditional distribution of the
linear form $L_2=\beta_1\xi_1+\beta_2\xi_2$ given
$L_1=\alpha_1\xi_1+\alpha_2\xi_2$ is symmetric. We describe all
possible distributions of $\mu_j.$ This is a group-theoretic analogue
of the well-known Heyde characterization of a Gaussian
distribution on the real line.



characterization theorem, discrete Abelian group, Heyde theorem

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Journal of the Austral. Maths Soc, copyright Australian Mathematical Society.