The Heyde theorem on a-adic solenoids

Abstract

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 .

On a Characterization Theorem for Probability Distributions on Discrete Abelian Groups

“…It is obvious that (25) implies the inclusion (I −α)(E 2 ) ⊂ E 2 . Hence, (20) holds, and then (21) follows from (17) and (19). Since group X contains no elements of order 2 and K 1 is a finite group, we have (K 1 ) (2) = K 1 .…”

On a characterization of convolutions of Gaussian and Haar distributions

“…Other generalizations of the Heyde theorem on groups were obtained in 3–5, 9, 17 (see also 6, Chap. VI]).…”

“…Lemma 3.4 (17) Let X be a locally compact Abelian group. Let δ 1 , δ 2 be continuous homomorphisms of the group X .…”

On a characterization theorem on a-adic solenoids

“…To prove Theorem 2 we need such lemmas. Lemma 6 ( [16]). Let X be a locally compact Abelian group, α be a topological automorphism of X.…”