The Heyde Theorem on a Group $$\mathbb {R}^n \times D$$, Where D is a Discrete Abelian Group

Abstract

Heyde proved that the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form given another. The present article is devoted to a group analogue of the Heyde theorem. We describe distributions of independent random variables \(\xi _1\) and \(\xi _2\) with values in a group \(X=\mathbf {R}^n\times D\), where D is a discrete Abelian group, which are characterized by the symmetry of the conditional distribution of the linear form \(L_2 = \xi _1 + \delta \xi _2\) given \(L_1 = \xi _1 + \xi _2\), where \(\delta \) is a topological automorphism of X such that \(\mathrm{Ker}(I+\delta )=\{0\}\).