On a generalisation of the Skitovich–Darmois theorem for several linear forms on Abelian groups

  • Gennadiy FeldmanEmail author


A.M. Kagan introduced a class of distributions \(\mathcal {D}_{m, k}\) in \(\mathbb {R}^m\) and proved that if the joint distribution of m linear forms of n independent random variables belongs to the class \(\mathcal {D}_{m, m-1}\), then the random variables are Gaussian. A.M. Kagan’s theorem implies, in particular, the well-known Skitovich–Darmois theorem, where the Gaussian distribution on the real line is characterized by independence of two linear forms of n independent random variables. In the note we describe a wide class of locally compact Abelian groups where A.M. Kagan’s theorem is valid.


Locally compact Abelian group Gaussian distribution Linear forms 

Mathematics Subject Classification

43A25 43A35 60B15 62E10 



The author thanks the reviewer for carefully reading the article and the language corrections.


  1. 1.
    Feldman, G.M.: Gaussian distributions on locally compact abelian groups. Theory Probab. Appl. 23(3), 529–542 (1979)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Feldman, G.M.: Bernstein Gaussian distributions on groups. Theory Probab. Appl. 31(1), 40–49 (1987)CrossRefGoogle Scholar
  3. 3.
    Feldman, G.M.: Marcinkiewicz and Lukacs theorems on abelian groups. Theory Probab. Appl. 34(2), 290–297 (1990)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Feldman, G.M.: Characterization of the Gaussian distribution on groups by the independence of linear statistics. Sib. Math. J. 31(2), 336–345 (1990)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Feldman, G.M.: On the Skitovich–Darmois theorem on compact groups. Theory Probab. Appl. 41(4), 768–773 (1997)MathSciNetGoogle Scholar
  6. 6.
    Feldman, G.M.: The Skitovich–Darmois theorem for discrete periodic Abelian groups. Theory Probab. Appl. 42(4), 611–617 (1998)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Feldman, G.M.: A characterization of the Gaussian distribution on Abelian groups. Probab. Theory Relat. Fields. 126(1), 91–102 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Feldman, G.M.: Functional Equations and Characterization Problems on Locally Compact Abelian Groups. EMS Tracts in Mathematics, vol. 5. European Mathematical Society (EMS), Zurich (2008)Google Scholar
  9. 9.
    Feldman, G.M.: On a theorem of K. Schmidt. Bull. Lond. Math. Soc. 41(1), 103–108 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Feldman, G.M.: Characterization theorems for \(Q\)-independent random variables with values in a locally compact Abelian group. Aequ. Math. 91, 949–967 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Feldman, G.M., Graczyk, P.: The Skitovich–Darmois theorem for locally compact Abelian groups. J. Aust. Math. Soc. 88(3), 339–352 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Feldman, G.M., Myronyuk, M.V.: Independent linear forms on connected Abelian groups. Math. Nachr. 284(2–3), 255–265 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis, vol. 1. Springer, Berlin (1963)zbMATHCrossRefGoogle Scholar
  14. 14.
    Kagan, A.M.: New classes of dependent random variables and a generalization of the Darmois–Skitovich theorem to several forms. Theory Probab. Appl. 33(2), 286–295 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Kagan, A.M., Linnik, YuV, Rao, C.R.: Characterization Problems in Mathematical Statistics, Wiley Series in Probability and Mathematical Statistics. Wiley, New York (1973)Google Scholar
  16. 16.
    Kagan, A.M., Székely, G.J.: An analytic generalization of independence and identcal distributiveness. Stat. Probab. Lett. 110, 244–248 (2016)zbMATHCrossRefGoogle Scholar
  17. 17.
    Lisyanoi, S.V.: On a group analogue of a theorem of A. M. Kagan. Theory Probab. Appl. 40(1), 165–167 (1995)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Myronyuk, M.V., Feldman, G.M.: Independent linear statistics on the two-dimensional torus. Theory Probab. Appl. 52(1), 78–92 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Parthasarathy, K.R.: Probability Measures on Metric Spaces. Academic Press, New York (1967)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of UkraineKharkivUkraine

Personalised recommendations