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Domains of attraction for the subcritical Galton-Watson branching process

Published online by Cambridge University Press:  14 July 2016

H. Rubin  and
D. Vere-Jones
H. Rubin
Affiliation:
Michigan State University
D. Vere-Jones
Affiliation:
Australian National University
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Extract

Let F(z) = σ fjzj be the generating function for the offspring distribution {fj} from a single ancestor in the usual Galton-Watson process. It is well-known (see Harris [1]) that if Π(z) is the generating function of the distribution of ancestors in the 0th generation, the distribution of offspring at the nth generation has generating function where Fn(z), the nth functional iterate of F(z), gives the distribution of offspring at the nth generation from a single ancestor.

Type
Short Communications
Information
Journal of Applied Probability , Volume 5 , Issue 1 , April 1968 , pp. 216 - 219
Copyright
Copyright © Sheffield: Applied Probability Trust 

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References

[1] Harris, T. E. (1963) The Theory of Branching Processes. Springer, Berlin.Google Scholar
[2] Heathcote, C. R., Seneta, E., and Vere-Jones, D. (1967) A refinement of two theorems in the theory of branching processes. Teor. Verojat. Primen. 12, 341–346.Google Scholar
[3] Seneta, E. and Vere-Jones, D. (1966) On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states. J. Appl. Prob. 3, 403–34.Google Scholar