On Heyde’s Theorem for Probability Distributions on a Discrete Abelian Group¹

Abstract—Let X be a countable discrete Abelian group containing no elements of order 2. Let $\alpha $ be an automorphism of X. Let ${{\xi }_{1}}$ and ${{\xi }_{2}}$ be independent random variables with values in the group X and distributions ${{\mu }_{1}}$ and ${{\mu }_{2}}$. The main result of the article is the following statement. The symmetry of the conditional distribution of the linear form ${{L}_{2}} = {{\xi }_{1}} + \alpha {{\xi }_{2}}$ given ${{L}_{1}} = {{\xi }_{1}} + {{\xi }_{2}}$ implies that ${{\mu }_{j}}$ are shifts of the Haar distribution of a finite subgroup of X if and only if $\alpha $ satisfies the condition ${\text{Ker}}(I + \alpha )$ = {0}. Some generalisations of this theorem are also proved.