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On the Skitovich-Darmois Theorem on Abelian Groups

Ukrainian Mathematical Journal volume 56, pages 1602–1618 (2004)Cite this article

Abstract

We prove theorems that generalize the Skitovich-Darmois theorem to the case where independent random variables ξj, j = 1, 2, ..., n, n ≥ 2, take values in a locally compact Abelian group and the coefficients αj and βj of the linear forms L 1 = α1ξ1 + ... + αnξn and L 2 = β1ξ1 + ... + βnξn are automorphisms of this group.

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REFERENCES

  1. V. P. Skitovich, “On one property of normal distribution,” Dokl. Akad. Nauk SSSR, 89, 217–219 (1953).

    Google Scholar 

  2. G. Darmois, “Analyse generale des liaisons stochastiques,” Rev. Inst. Int. Statist., 21, 2–8 (1953).

    Google Scholar 

  3. A. M. Kagan, Yu. V. Linnik, and S. R. Rao, Characterization Problems in Mathematical Statistics [in Russian], Nauka, Moscow (1972).

    Google Scholar 

  4. S. G. Ghurye and I. Olkin, “A characterization of the multivariate normal distribution,” Ann. Math. Statist., 33, 533–541 (1962).

    Google Scholar 

  5. D. Neuenschwander, B. Roynette, and R. Schott, “Characterization of Gauss measures on nilpotent Lie groups and symmetric spaces,” C. R. Acad. Sci., Ser. I, 324, 87–92 (1997).

    Google Scholar 

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

    Google Scholar 

  7. G. M. Fel’dman, “On the Skitovich-Darmois theorem on Abelian groups,” Teor. Ver. Primen., 37, No.4, 695–708 (1992).

    Google Scholar 

  8. G. M. Fel’dman, “Skitovich-Darmois theorem for compact groups,” Teor. Ver. Primen., 41, No.4, 901–906 (1996).

    Google Scholar 

  9. G. M. Fel’dman, “Skitovich-Darmois theorem for discrete periodic Abelian groups,” Teor. Ver. Primen., 42, No.4, 747–756 (1997).

    Google Scholar 

  10. G. M. Fel’dman, “On the Skitovich-Darmois theorem for finite Abelian groups,” Teor. Ver. Primen., 45, No.3, 603–607 (2000).

    Google Scholar 

  11. G. M. Fel’dman and P. Graczyk, “On the Skitovich-Darmois theorem on compact Abelian groups,” J. Theor. Probab., 13, No.3, 859–869 (2000).

    Article  Google Scholar 

  12. P. Graczyk and G. M. Fel’dman, “Independent linear statistics on finite Abelian groups,” Ukr. Mat. Zh., 53, No.4, 441–448 (2001).

    Google Scholar 

  13. G. M. Fel’dman and P. Graczyk, “On the Skitovich-Darmois theorem for discrete Abelian groups,” in: Univ. Angers. Prepubl. Dep. Math., No. 155 (2002).

  14. G. M. Fel’dman, “A characterization of Gaussian distribution on Abelian groups,” Probab. Theor. Relat. Fields, 126, No.1, 91–102 (2003).

    Article  Google Scholar 

  15. K. R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, New York (1967).

    Google Scholar 

  16. E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vol. 1, Springer, Berlin (1963).

    Google Scholar 

  17. L. Fuchs, Infinite Abelian Groups, Vol. I, Academic Press, New York (1970).

    Google Scholar 

  18. L. Fuchs, Infinite Abelian Groups, Vol. II, Academic Press, New York (1973).

    Google Scholar 

  19. G. M. Fel’dman, “On decomposition of Gaussian distribution on groups,” Teor. Ver. Primen., 22, No.1, 136–143 (1977).

    Google Scholar 

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Authors and Affiliations

  1. Physicotechnical Institute of Low Temperatures, Ukrainian Academy of Sciences, Kharkiv, Ukraine

    M. V. Myronyuk

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  1. M. V. Myronyuk
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 10, pp. 1342 – 1356, October, 2004.

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Myronyuk, M.V. On the Skitovich-Darmois Theorem on Abelian Groups. Ukr Math J 56, 1602–1618 (2004). https://doi.org/10.1007/s11253-005-0137-3

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  • DOI: https://doi.org/10.1007/s11253-005-0137-3

Keywords

  • Abelian Group
  • Linear Form
  • Independent Random Variable
  • Compact Abelian Group