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A characterization of the Gaussian distribution on Abelian groups
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  • Published: May 2003

A characterization of the Gaussian distribution on Abelian groups

  • G.M. Feldman1 

Probability Theory and Related Fields volume 126, pages 91–102 (2003)Cite this article

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Abstract.

 It is well known that the independence of two linear forms with nonzero coefficients of independent random variables implies that the random variables are Gaussian (the Skitovich-Darmois theorem). The analogous result holds true for two linear forms of independent random vectors with nonsingular matrices as coefficients (the Ghurye-Olkin theorem). In this paper we give the complete description of locally compact Abelian groups X for which the independence of two linear forms of independent random variables with values in X having distributions with nonvanishing characteristic functions (coefficients of the forms are topological automorphisms of X) implies that the random variables are Gaussian.

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Authors and Affiliations

  1. Mathematical Division, Institute for Low Temperature Physics and Engineering, 47, Lenin ave, Kharkov, 61103, Ukraine. e-mail: feldman@ilt.kharkov.ua, , , , , , UA

    G.M. Feldman

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  1. G.M. Feldman
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Additional information

Received: 24 July 2001 / Revised version: 19 January 2003 Published online: 28 March 2003

Mathematics Subject Classification (2000): primary 62E10; secondary 60B15

Key words or phrases: Gaussian distribution – Skitovich-Darmois theorem – Locally compact Abelian group

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Feldman, G. A characterization of the Gaussian distribution on Abelian groups. Probab. Theory Relat. Fields 126, 91–102 (2003). https://doi.org/10.1007/s00440-003-0256-4

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  • Issue Date: May 2003

  • DOI: https://doi.org/10.1007/s00440-003-0256-4

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Keywords

  • Gaussian Distribution
  • Characteristic Function
  • Abelian Group
  • Linear Form
  • Random Vector
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