On the Skitovich–Darmois Theorem for Compact Abelian Groups

G. M. Feldman and P. Graczyk
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G. M. Feldman: Institute for Low Temperature Physics and Engineering
P. Graczyk: Université d'Angers

Journal of Theoretical Probability, 2000, vol. 13, issue 3, 859-869

Abstract: Abstract Let X be a separable compact Abelian group, Aut(X) the group of topological automorphisms of X, f n: X→X a homomorphism f n(x)=nx, and X (n)=Im f n. Denote by I(X) the set of idempotent distributions on X and by Γ(X) the set of Gaussian distributions on X. Consider linear statistics L 1=α 1(ξ 1)+α 2(ξ 2) and L 2=β 1(ξ 1)+β 2(ξ 2), where ξ j are independent random variables taking on values in X and with distributions μ j, and α j, β j∈Aut(X). The following results are obtained. Let X be a totally disconnected group. Then the independence of L 1 and L 2 implies that μ 1, μ 2∈I(X) if and only if X possesses the property: for each prime p the factor-group X/X (p) is finite. If X is connected, then there exist independent random variables ξ j taking on values in X and with distributions μ j, and α j, β j∈Aut(X) such that L 1 and L 2 are independent, whereas μ 1, μ 2∉Γ(X) * I(X).

Keywords: characterization of probability distributions; independence of linear statistics; compact Abelian group (search for similar items in EconPapers)
Date: 2000
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Citations: View citations in EconPapers (3)

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DOI: 10.1023/A:1007870814570

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