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On the Skitovich–Darmois Theorem for Compact Abelian Groups

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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).

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Authors and Affiliations

  1. Institute for Low Temperature Physics and Engineering, 47, Lenin ave, Kharkov, 310164, Ukraine

    G. M. Feldman

  2. Université d'Angers, 2 boulevard Lavoisier, 49045, Angers, France

    P. Graczyk

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  1. G. M. Feldman
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  2. P. Graczyk
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Feldman, G.M., Graczyk, P. On the Skitovich–Darmois Theorem for Compact Abelian Groups. Journal of Theoretical Probability 13, 859–869 (2000). https://doi.org/10.1023/A:1007870814570

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