Generalization of the Heyde Theorem to Some Locally Compact Abelian Groups

On a Characterization Theorem for Locally Compact Abelian Groups Containing an Element of Order 2

“…Assume σ 1 > 0, σ 2 > 0. Using Lemma 2.8 and ( 15) and reasoning as in the proof of Lemma 3.3 in [11], we make sure that for each fixed n ∈ Z(2), h ∈ H, the functions μj (s, n, h) can be extended to the complex plane C as entire functions in s, equation ( 11) holds for all s j ∈ C, n j ∈ Z(2), h j ∈ H and the functions μj (s, n, h) do not vanish for all s ∈ C, n ∈ Z(2), h ∈ H.…”

“…Substitute s 1 = s 2 = 0 and n 1 = n 2 = 0 in equation (10). Taking into account Lemma 2.1 and reasoning as in the proof of Theorem 3.1 in [11], it follows from the obtained equation and Lemma 2.2 that there exist elements g 1 , g 2 ∈ G such that if θ j = µ j * E −g j and η j are independent random variables with values in the group X and distributions θ j , then the conditional distribution of the linear form…”

“…By the condition of the theorem, G is a finite Abelian group containing no elements of order 2. Hence H (2) = H. Substitute representations (16) for the characteristic functions μj (s, n, h) into equation (11). Put n 1 = n 2 = 0, h 1 = h 2 = h in the obtained equation.…”

“…Substitute representations (16) for the characteristic functions μj (s, n, h) into equation (11). Setting now n 1 = 1, n 2 = 0, h 1 = h 2 = h in the obtained equation and taking into account that H (2) = H, we get…”

“…In article [11] Heyde's theorem for two independent random variables was generalized to groups of the form R × G, where G is a finite Abelian group containing no elements of order 2. The presence of elements of order 2 in a group plays an exceptional role in generalizing Heyde's theorem.…”