Reduced Branching Processes in Random Environment

  • Vatutin Vladimir
  • Dyakonova Elena
Conference paper
Part of the Trends in Mathematics book series (TM)


Let Zn be the number of particles at time n in a critical branching process in random environment and Zm,n be the number of particles in this process at time m ≤ n which have non-empty offspring at time n. We prove limit theorems for the process {Z[nt],n, t ∈ (0, 1]} conditioned on the event {Zn > 0}. Quenched and annealed approaches are considered.*


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    B.K.Athreya and P.Ney, (1972) Branching processes, Springer-Verlag, Berlin.zbMATHCrossRefGoogle Scholar
  2. [2]
    K.A.Borovkov and V.A.Vatutin, (1997) Reduced critical branching processes in random environment, Stoch. Proc. Appl., 71, No.2, 225–240.MathSciNetzbMATHGoogle Scholar
  3. [3]
    J.Geiger and G.Kersting, (2000) The survival probability of a critical branching process in random environment, Theory Probab. Appl., 45, No. 3, 607–615.MathSciNetGoogle Scholar
  4. [4]
    K.Fleischmann and U.Prehn, (1974) Ein Grenzfersatz für subkritische Verzweigungsprozesse mit eindlich vielen Typen von Teilchen, Math. Nachr. 64, 233–241.MathSciNetCrossRefGoogle Scholar
  5. [5]
    K.Fleischmann and R.Siegmund-Schultze, (1977) The structure of reduced critical Galton-Watson processes, Math. Nachr. 79, 357–362.Google Scholar
  6. [6]
    K.Fleischmann and V.A.Vatutin, (1999) Reduced subcritical branching processes in random environment, Adv. Appl. Probab., 31, No.1, 88–111.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    F.Spitzer, (1964) Principles of random walk, Princeton, New Jersey.zbMATHGoogle Scholar
  8. [8]
    V.A.Vatutin and A.M. Zubkov, (1993) Branching Processes II., Journal of Soviet Mathematics, 67, 3407–3485.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    V.A.Vatutin, (2002) Reduced branching processes in a random environment: the critical case, Theory Probab. Appl., 47, No.1, 21--38 (In Russian.).MathSciNetGoogle Scholar
  10. [10]
    A.M.Zubkov, (1975) Limit distributions of the distance to the closest mutual ancestor, Theory Probab. Appl., 20, No. 3, 602–612.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Basel AG 2002

Authors and Affiliations

  • Vatutin Vladimir
    • 1
  • Dyakonova Elena
    • 1
  1. 1.Steklov Mathematicla InstituteGSP-1 MoscowRussia

Personalised recommendations