By the classical Skitovich-Darmois Theorem the independence of two linear forms of independent random variables characterizes a Gaussian distribution. A result close to the Skitovich-Darmois Theorem was proved by Heyde, with the condition of the independence of linear forms replaced by the symmetry of the conditional distribution of one linear form given the other. The present article is devoted to an analog of Heyde's Theorem in the case when random variables take values in a finite Abelian group and the coefficients of the linear forms are group automorphisms.
Let X be a countable discrete abelian group with automorphism group Aut(X ). Let ξ 1 and ξ 2 be independent X -valued random variables with distributions µ 1 and µ 2 , respectively. Suppose that α 1 , α 2 , β 1 , β 2 ∈ Aut(X ) andAssuming that the conditional distribution of the linear form L 2 given L 1 is symmetric, where L 2 = β 1 ξ 1 + β 2 ξ 2 and L 1 = α 1 ξ 1 + α 2 ξ 2 , we describe all possibilities for the µ j . This is a group-theoretic analogue of Heyde's characterization of Gaussian distributions on the real line.2000 Mathematics subject classification: primary 60B15; secondary 62E10.
We prove the following analogue of the Heyde theorem for a-adic solenoids. Let ξ 1 , ξ 2 be independent random variables taking values in an a-adic solenoid Σ a and with distributions µ 1 , µ 2 . Let α j , β j be topological automorphisms of Σ a such that β 1 α −1 1 ± β 2 α −1 2 are topological automorphisms of Σ a too. Assuming that the conditional distribution of the linear form L 2 = β 1 ξ 1 + β 2 ξ 2 given L 1 = α 1 ξ 1 + α 2 ξ 2 is symmetric, we describe possible distributions µ 1 , µ 2 .
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