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On the Skitovich–Darmois Theorem for Compact Abelian Groups

Author

Listed:
  • G. M. Feldman

    (Institute for Low Temperature Physics and Engineering)

  • P. Graczyk

    (Université d'Angers)

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).

Suggested Citation

  • G. M. Feldman & P. Graczyk, 2000. "On the Skitovich–Darmois Theorem for Compact Abelian Groups," Journal of Theoretical Probability, Springer, vol. 13(3), pages 859-869, July.
    • Handle: RePEc:spr:jotpro:v:13:y:2000:i:3:d:10.1023_a:1007870814570
    DOI: 10.1023/A:1007870814570
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    Citations

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    Cited by:

    1. G. M. Feldman, 2004. "On the Heyde Theorem for Finite Abelian Groups," Journal of Theoretical Probability, Springer, vol. 17(4), pages 929-941, October.
    2. Margaryta Myronyuk, 2020. "Independent Linear Forms on the Group $$\varOmega _p$$Ωp," Journal of Theoretical Probability, Springer, vol. 33(1), pages 1-21, March.
    3. Gennadiy Feldman, 2015. "On the Skitovich–Darmois Theorem for the Group of $$p$$ p -Adic Numbers," Journal of Theoretical Probability, Springer, vol. 28(2), pages 539-549, June.

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