Stochastic Monotonicity and Branching Processes

  • Harry Cohn
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 84)


It is shown that stochastic monotonicity provides a unified treatment of simple and multitype branching processes. In the supercritical case the key to con­vergence in probability or a.s. of suitably normed branching processes is a law of large numbers for some independent copies of random variables. Applications to branching processes in varying environment are given.


Independent Copy Supercritical Case Offspring Distribution Branching Process Dependent Offspring 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Aldous, D., Tail behavior of birth-and-death and stochastically monotone se­quences, Z. Wahrscheinlich. Verw. Gebiete 62 (1983) pp. 375–394.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    Asmussen S. and Hering H. Branching Processes Birkhauser, Boston, (1983).Google Scholar
  3. [3]
    Athreya K.B. and Ney P. Branching Processes Springer-Verlag New York (1972).Google Scholar
  4. [4]
    Biggins, J., Cohn, H., and Nerman, O., Multitype branching processes in varying environment In preparation.Google Scholar
  5. [5]
    Cohn H. On the convergence of stochastically monotone sequences of random variables and some applications J. Appl. Probab. 18 (1981) pp. 592_605.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    Cohn H. On a property related to convergence in probability and some applica­tions to branching processes Stoch. Proc. Appl. 12 (1982) pp. 59–72.CrossRefGoogle Scholar
  7. [7]
    Cohn H. On the fluctuation of stochastically monotone Markov chains and ap­plications J. Appl. Probab. 20 (1983) pp. 178_184.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    Cohn H. Limit behaviour for stochastic monotonicity and applications Adv. Appl. Probab. 20 (1988) pp. 331–347.zbMATHCrossRefGoogle Scholar
  9. [9]
    Daley D.J. Stochastically monotone Markov chains, Z. Wahrscheinlich, 10 (1968), pp. 305–317.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    Jagers, P., Branching Processes with Biological Applications, Wiley, Lon­don, (1975). zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Harry Cohn
    • 1
  1. 1.Department of StatisticsMelbourne UniversityParkvilleAustralia

Personalised recommendations