### Heyde's characterisation theorem for discrete Abelian Groups

#### Abstract

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.

doi:10.1017/S1446788709000378

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.

doi:10.1017/S1446788709000378

#### Keywords

characterization theorem, discrete Abelian group, Heyde theorem

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