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On a characterization theorem on finite Abelian groups


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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  1. Kagan A. M., Linnik Yu. V., and Rao S. R., Characterization Problems in Mathematical Statistics, Wiley, New York (1973).

    Google Scholar 

  2. Heyde C. C., “Characterization of the normal law by the symmetry of a certain conditional distribution,” Sankhya. Ser. A, 32, 115–118 (1970).

    Google Scholar 

  3. Rukhin A. L., “On a theorem of S. N. Bernstein,” Math. Notes, 6, 638–641 (1969).

    Google Scholar 

  4. Heyer H. and Rall Ch., “Gausssche Wahrscheinlichkeitsmasse auf Corwinschen Gruppen,” Math. Z., Bd 128, 343–361 (1972).

    Google Scholar 

  5. Feldman G. M., “Bernstein Gaussian distributions on groups,” Theory Probab. Appl., 31, No.1, 40–49 (1987).

    Google Scholar 

  6. Neuenschwander D., “Gauss measures in the sense of Bernstein on the Heisenberg group,” Probab. Math. Statist., 14, No.2, 253–256 (1993).

    Google Scholar 

  7. Neuenschwander D., Roynette B., and Schott R., “Characterizations of Gaussian distributions on simply connected nilpotent Lie groups and symmetric spaces,” C. R. Acad. Sci. Paris. Sèr. I., 324, 87–92 (1997).

    Google Scholar 

  8. Neuenschwander D. and Schott R., “The Bernstein and Skitovich-Darmois characterization theorems for Gaussian distributions on groups, symmetric spaces, and quantum groups,” Exposition Math., 15, 289–314 (1997).

    Google Scholar 

  9. Feldman G. M. and Graczyk P., “On the Skitovich-Darmois theorem on compact Abelian groups,” J. Theoret. Probab., 13, 859–869 (2000).

    Google Scholar 

  10. Graczyk P. and Loeb J.-J., “A Bernstein property of measures on groups and symmetric spaces,” Probab. Math. Statist., 20, No.1, 141–149 (2000).

    Google Scholar 

  11. Feldman G. M., “A characterization of the Gaussian distribution on Abelian groups,” Probab. Theory Related Fields, 126, 91–102 (2003).

    Google Scholar 

  12. Feldman G. M., “On the Heyde theorem for finite Abelian groups,” J. Theoret. Probab., 17, 929–941 (2004).

    Google Scholar 

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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).

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