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The Galton-Watson process with mean one

Published online by Cambridge University Press:  14 July 2016

E. Seneta
E. Seneta*
Affiliation:
Australian National University
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Extract

Let Zn be the number of individuals in the n-th generation of a simple (discrete, one-type) Galton-Watson process, descended from a single ancestor. Write and assume, as usual, that 0 < F(0) < 1. It is well known that the p.g.f. of Zn is Fn(s), the n-th functional iterate of F(s), and that if the process is construed as a Markov chain on the states 0, 1, 2,…, then its n-step transition probabilities are given by

Type
Research Papers
Information
Journal of Applied Probability , Volume 4 , Issue 3 , November 1967 , pp. 489 - 495
Copyright
Copyright © Applied Probability Trust 

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References

[1] Breny, H. (1962) Sur un point de la théorie des files d'attente. Ann. Soc. Sci. Bruxelles. 76, 5–12 Google Scholar
[2] Feller, W. (1957) An Introduction to Probability Theory and its Applications. I. Wiley, New York.Google Scholar
[3] Harris, T. E. (1963) The Theory of Branching Processes. Springer, Berlin.Google Scholar
[4] Kesten, H., Ney, P. and Spitzer, F. (1966) The Galton-Watson Process with mean one and finite variance. Teor. Veroyat. Primen, 11, 579–611.Google Scholar
[5] Kuczma, M. (1965) On convex solutions of Abel's functional equation. Bull. Acad. Polon. Sci. 13, 645–648.Google Scholar
[6] Lévy, P. (1928) Fonctions à croissance régulière et itération d'ordre fractionnaire. Ann. Mat. Pura Applic. (4) 5, 269–298.CrossRefGoogle Scholar
[7] 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, 403–434.Google Scholar
[8] Sekeres, G. (1960) On a theorem of Paul Lévy. Magyar Tud. Akad. Mat. Kutató Int. Közl. A, 5, 277–282.Google Scholar