The Heyde theorem on a group ℝ^n× D, where D is a discrete Abelian group

07/23/2020

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by   Margaryta Myronyuk, et al.

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Heyde proved that a Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear statistic given another. The present article is devoted to a group analogue of the Heyde theorem. We describe distributions of independent random variables ξ_1, ξ_2 with values in a group X=ℝ^n× D, where D is a discrete Abelian group, which are characterized by the symmetry of the conditional distribution of the linear statistic L_2 = ξ_1 + δξ_2 given L_1 = ξ_1 + ξ_2, where δ is a topological automorphism of X such that Ker(I+δ)={0}.