On a characterization theorem for locally compact abelian groups

Abstract

The well-known Skitovich-Darmois theorem asserts that a Gaussian distribution is characterized by the independence of two linear forms of independent random variables. The similar result was proved by Heyde, where instead of the independence, the symmetry of the conditional distribution of one linear form given another was considered. In this article we prove that the Heyde theorem on a locally compact Abelian group X remains true if and only if X contains no elements of order two. We describe also all distributions on the two-dimensional torus X = T 2 which are characterized by the symmetry of the conditional distribution of one linear form given another. In so doing we assume that the coefficients of the forms are topological automorphisms of X and the characteristic functions of the considering random variables do not vanish.

On the Skitovich-Darmois theorem and Heyde theorem in a Banach space

“…As in the proof of Theorem 1, we show that the proof of Theorem 2 reduces to the case where the linear forms have the form L 1 = ξ 1 + … + ξ n and L 2 = C 1 1 ξ + … + C n n ξ , where C X j ∈Aut ( ) are such that C C i j ± ∈ Aut ( ) X for all i ≠ j, and the characteristic functionals satisfy the inequality ˆ( ) μ j f ≥ 0. As in [4], we show that the condition of the symmetry of the conditional distribution of L 2 for fixed L 1 is equivalent to the statement that the characteristic functionals ˆ( ) μ j f satisfy the equation…”

The Heyde theorem for locally compact Abelian groups

“…It was proved in [4] that the symmetry of the conditional distribution of L 2 given L 1 implies that all μ j ∈ Γ (X) if and only if X contains no elements of order 2. If a group X contains elements of order 2, then the following natural problem arises: Problem 1.…”

“…This problem was solved in [4] for the case when X is the two-dimensional torus T 2 . Namely, the following theorem holds: Let ξ 1 , ξ 2 be independent random variables with values in T 2 and distributions μ j with non-vanishing characteristic functions.…”

On a Characterization of the Haar Distribution on Compact Abelian Groups

“…The case when random variables take values in locally compact Abelian groups was studied in details (see, e.g., [1][2][3][4][5][6][7][8], ).…”