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Self-decomposable discrete distributions and branching processes

  • K. van Harn
  • F. W. Steutel
  • W. Vervaat
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 861)

Abstract

Self-decomposable distributions are known to be absolutely continuous. In this note analogues of the concept of self-decomposability are proposed for distributions on the set ℕ0 of nonnegative integers. To each of them corresponds an analogue of multiplication (in distribution) that preserves ℕ0-valuedness and is characterized by a composition semigroup of probability generating functions, such as occur in branching processes.

Keywords

Nonnegative Integer Absolute Continuity Continuous Semigroup Probability Generate Function Divisible Distribution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Athreya, K.B. & Ney, P.E., Branching processes. Springer, Berlin 1972.CrossRefzbMATHGoogle Scholar
  2. [2]
    Fisz, M. & Varadarajan, V.S., A condition for absolute continuity of infinitely divisible distribution functions. Z. Wahrscheinlichkeitstheorie verw. Gebiete 1 (1963), 335–339.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Forst, G., A characterization of self-decomposable probabilities on the half-line. Z. Wahrscheinlichkeitstheorie verw. Gebiete 49 (1979), 349–352.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    van Harn, K., Classifying infinitely divisible distributions by functional equations. Math. Centre Tracts 103, Math. Centre, Amsterdam 1978.zbMATHGoogle Scholar
  5. [5]
    Harris, T.E., The theory of branching processes. Springer, Berlin 1963.CrossRefzbMATHGoogle Scholar
  6. [6]
    Steutel, F.W. & van Harn, K., Discrete analogues of self-decomposability and stability. Ann. Probability 7 (1979), 893–899.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Steutel, F.W., Vervaat, W. & Wolfe, S.J., Integer-valued branching processes with immigration. Forthcoming.Google Scholar

Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • K. van Harn
    • 1
  • F. W. Steutel
    • 2
  • W. Vervaat
    • 3
  1. 1.Wiskundig SeminariumVrije UniversiteitAmsterdam
  2. 2.Onderafd. der Wiskunde Techn. HogeschoolEindhoven
  3. 3.Math. InstituutKath. UniversiteitNijmegen

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