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