On a characterization of shifts of Haar distributions on compact open subgroups of a compact Abelian group

Abstract

Let X be a compact Abelian group. In the article we obtain a characterization of shifts of Haar distributions on compact open subgroups of the group X by the symmetry of the conditional distribution of one linear form of independent random variables taking values in X given another. Coefficients of the linear forms are topological automorphisms of the group X. This result can be viewed as an analogue for compact Abelian groups of the well-known Heyde theorem, where the 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.

17 pages