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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 F n (z), the nth functional iterate of F(z), gives the distribution of offspring at the nth generation from a single ancestor.

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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