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On a characterization theorem on finite Abelian groups

Abstract

By the classical Skitovich-Darmois Theorem the independence of two linear forms of independent random variables characterizes a Gaussian distribution. A result close to the Skitovich-Darmois Theorem was proved by Heyde, with the condition of the independence of linear forms replaced by the symmetry of the conditional distribution of one linear form given the other. The present article is devoted to an analog of Heyde’s Theorem in the case when random variables take values in a finite Abelian group and the coefficients of the linear forms are group automorphisms.

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Original Russian Text Copyright © 2005 Myronyuk M. V. and Feldman G. M.

Translated from Sibirski \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\imath}\) Matematicheski \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\imath}\) Zhurnal, Vol. 46, No. 2, pp. 403–415, March–April, 2005.

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Myronyuk, M.V., Feldman, G.M. On a characterization theorem on finite Abelian groups. Sib Math J 46, 315–324 (2005). https://doi.org/10.1007/s11202-005-0033-y

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  • DOI: https://doi.org/10.1007/s11202-005-0033-y

Keywords

  • characterization of probability distributions
  • idempotent distributions
  • finite Abelian groups