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On asymptotic properties of sub-critical branching processes

Published online by Cambridge University Press:  09 April 2009

E. Seneta
E. Seneta
Affiliation:
Australian National UniversityCanberra
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Let Zn be the numer of individuals in the nth generation of a discrete branching process, descended from a single a singel ancestor, for which we put It is well known that the probability generating function of Zn is Fn(s), the n-th functional iterate of F(s), and that if m = EZ1 does not exceed unity, then lim (Harris [1], Chapter 1). In particular, extinction is certain.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 8 , Issue 4 , November 1968 , pp. 671 - 682
Copyright
Copyright © Australian Mathematical Society 1968

References

[1]Harris, T. E., The Theory of Branching Processes (Springer-Verlag, Berlin, 1963).CrossRefGoogle Scholar
[2]Heathcote, C. R. and Seneta, E., ‘Inequalities for branching processes’, J. Appl. Prob. 3 (1966), 261–7. (See also correction: 4 (1967), 215.)CrossRefGoogle Scholar
[3]Nagaev, S. V. and Muhamedhanova, R., ‘Transient phenomena in branching stochastic processes with discrete time’, (in Russian), Limit Theorems and Statistical Inference (pp. 83–9, Izd. ‘FAN’, Uzhbekskoi S.S.R. Tashkent, 1966).Google Scholar
[4]Seneta, E., ‘On the transient behaviour of a Poisson branching process’, J. Aust. Math. Soc. 7 (1967), 465–80.CrossRefGoogle Scholar
[5]Sevastyanov, B. A., ‘Transient phenomena in branching stochastic processes’, (in Russian). Teor. Veroiat. Primenen. 4 (1959), 121–35.Google Scholar