On a characterisation theorem for probability distributions on discrete Abelian groups

TL;DR

This paper characterizes probability distributions on certain discrete Abelian groups using symmetry conditions of linear forms, extending a classical theorem from real Gaussian distributions to a broader algebraic setting.

Contribution

It establishes a new characterization theorem for distributions on discrete Abelian groups, generalizing the Heyde theorem to non-Gaussian, algebraic contexts.

Findings

Symmetry of conditional distributions implies distributions are shifts of Haar measures.

The theorem holds if and only if the automorphism satisfies a specific kernel condition.

Generalizations of the main theorem are also provided.

Abstract

Let $X$ be a countable discrete Abelian group containing no elements of order 2, $\alpha$ be an automorphism of $X$, $\xi_1$ 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 the automorphism $\alpha$ satisfies the condition ${\rm Ker}(I+\alpha)=\{0\}$. This theorem is an analogue for discrete Abelian groups the well-known Heyde theorem where Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form of independent random variables given another. We also prove some generalisations of this theorem.