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Limiting Distributions in Branching Processes with Two Types of Particles

  • Michael Drmota
  • Vladimir Vatutin
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 84)

Abstract

Let us consider a decomposable e branching process with two types of particles T 1, T2 such that particles of type T 2 can only produce particles of types T 1 whereas particles of type T 1 can produce particles of both types. The aim of this paper is to characterize the kind of distribution of the particles of types T1 and T 2 when the total number n of all particles is fixed. Especially we are interested in the limit case n — ∞. It turns out that depending on the parameters of the process a number of different limiting distributions, e.g. normal or x 2 distributions, appear.

Key words

branching processes limiting distributions 

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References

  1. [Be]
    E.A. Bender, Central and local limit theorems applied to asymptotic enumeration, J. Combinatorial Th., Ser. A, 15, 91–111, 1973.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [Drl]
    M. Drmota, Asymptotic distributions and a multivariate Darboux method in enumeration problems, J. Combinatorial Th., Ser. A, 67, 169–184, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [Dr2]
    M. Drmota, A bivariate asymptotic expansion of coefficients of powers of generating functions, Europ. J. Combinatorics, 15, 139–152, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [Dr3]
    M. Drmota, The height distribution of leaves in rooted trees, Discr. Math. Appl., 4, 45–58, 1994.MathSciNetzbMATHGoogle Scholar
  5. [FO]
    PH. Flajolet AND A. Odlyzko, Singularity analysis of generating functions, SIAM J. Discrete Math., 3, 216–240, 1990.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [Sel]
    B.A. Sevastyanov, Verzweigungsprozesse, Akademie-Verlag, Berlin, 1974.Google Scholar
  7. [Se2]
    B.A. Sevastyanov, Final probabilities for branching sochastic processes,Theory Probab. Appl., 2, 133–135, 1957.MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Michael Drmota
    • 1
  • Vladimir Vatutin
    • 2
  1. 1.Department of Discrete MathematicsTechnical University of ViennaViennaAustria
  2. 2.Steklov Mathematical InstituteMoscow, GSP-1Russia

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