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

Abstract

Let X be a countable discrete Abelian group containing no elements of order 2. Let α be an automorphism of X. Let ξ₁ and ξ₂ be independent random variables with values in the group X and distributions μ₁ and μ₂. The main result of the article is the following statement. The symmetry of the conditional distribution of the linear form L₂ = ξ₁ + αξ₂ given L₁ = ξ₁ + ξ₂ implies that μ_j are shifts of the Haar distribution of a finite subgroup of X if and only if α satisfies the condition Ker(I + α)= {0}. Some generalisations of this theorem are also proved.