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On a characterization theorem for locally compact abelian groups
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  • Published: 15 March 2005

On a characterization theorem for locally compact abelian groups

  • G.M. Feldman1 

Probability Theory and Related Fields volume 133, pages 345–357 (2005)Cite this article

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Abstract.

The well-known Skitovich-Darmois theorem asserts that a Gaussian distribution is characterized by the independence of two linear forms of independent random variables. The similar result was proved by Heyde, where instead of the independence, the symmetry of the conditional distribution of one linear form given another was considered. In this article we prove that the Heyde theorem on a locally compact Abelian group X remains true if and only if X contains no elements of order two. We describe also all distributions on the two-dimensional torus which are characterized by the symmetry of the conditional distribution of one linear form given another. In so doing we assume that the coefficients of the forms are topological automorphisms of X and the characteristic functions of the considering random variables do not vanish.

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References

  1. Darmois, G.: Analyse generale des liaisons stochastiques. Rev. Inst. Intern. Stat. 21, 2–8 (1953)

    Google Scholar 

  2. Feldman, G.M.: On the decomposition of Gaussian distributions on groups. Theory Probab. Appl. 22, 133–140 (1977)

    Article  Google Scholar 

  3. Feldman, G.M.: Marcinkiewicz and Lukacs theorems on Abelian groups. Theory Probab. Appl. 34, 290-297 (1989).

    Article  Google Scholar 

  4. Feldman, G.M.: Arithmetic of probability distributions and characterization problems on Abelian groups. AMS translation of mathematical monographs 116, Providence, RI, 1993

  5. Feldman, G.M.: More on the Skitovich-Darmois theorem for finite Abelian groups. Theory Probab. Appl. 45, 507–511 (2001)

    Article  Google Scholar 

  6. Feldman, G.M.: A characterization of the Gaussian distribution on Abelian groups. Probab. Theory Relat. Fields. 126, 91–102 (2003)

    Article  Google Scholar 

  7. Feldman, G.M., Graczyk, P.: On the Skitovich-Darmois theorem on compact Abelian groups. J. of Theoretical Probability. 13, 859–869 (2000)

    Article  Google Scholar 

  8. Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis. 1. Springer-Verlag, Berlin Gottingen Heildelberg, 1963

  9. Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis. 2. Springer-Verlag, Berlin Heildelberg New York, 1970

  10. Heyde, C.C.: Characterization of the normal low by the symmetry of a certain conditional distribution. Sankhya, Ser. A. 31, 115–118 (1969)

    Google Scholar 

  11. Kagan, A.M., Linnik, Ju. V., Rao, C.R.: Characterization problems of mathematical statistics. Wiley, New York 1973

  12. Linnik Ju.V., Ostrovskii I.V.: Decomposition of random variables and vectors. AMS translation of mathematical monographs 48, Providence, RI, 1977

  13. Neuenschwander, D. and Schott, R.: The Bernstein and Skitovic-Darmois characterization theorems for Gaussian distributions on groups, symmetric spaces, and quantum groups. Expo. Math. 15, 289-314 (1997)

    Google Scholar 

  14. Parthasarathy, K.R.: Probability measures on metric spaces. Academic Press, New York London 1967

  15. Skitovich, V.P.: On a property of a normal distribution. Doklady Academii nauk SSSR 89, 217–219 (in Russian) (1953)

    Google Scholar 

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Authors and Affiliations

  1. Mathematical Division, B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, 47, Lenin ave, Kharkov, 61103, Ukraine

    G.M. Feldman

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  1. G.M. Feldman
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Correspondence to G.M. Feldman.

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Feldman, G. On a characterization theorem for locally compact abelian groups. Probab. Theory Relat. Fields 133, 345–357 (2005). https://doi.org/10.1007/s00440-005-0429-4

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  • Received: 23 October 2004

  • Revised: 18 December 2004

  • Published: 15 March 2005

  • Issue Date: November 2005

  • DOI: https://doi.org/10.1007/s00440-005-0429-4

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Keywords

  • Heyde theorem
  • Locally compact Abelian group
  • Characterization of probability distributions

Mathematics Subject Classifications (2000)

  • Primary 62E10
  • secondary 60B15
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