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The Galton-Watson process with infinite mean

Published online by Cambridge University Press:  14 July 2016

D. A. Darling
D. A. Darling*
Affiliation:
University of California, Irvine
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Extract

Let Zn be the number of descendents in the nth generation of a simple Galton-Watson branching process, initiated by a single progenitor, Z0 = 1. If E(Z1) < ∞ the limiting distribution of Zn is known in some detail, and a comprehensive account is given in Seneta [1]. If E{Z1) = ∞ (the “explosive” case) the behavior of the distribution of Zn for large n seems not to be known. As shown by Seneta [1], there are no constants cn such that cnZn has a non-degenerate limiting distribution but it turns out that, under conditions given below, log(Zn + 1) has a limiting distribution.

Type
Short Communications
Information
Journal of Applied Probability , Volume 7 , Issue 2 , August 1970 , pp. 455 - 456
Copyright
Copyright © Applied Probability Trust 1970 

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References

[1] Seneta, E. (1969) Functional equations and the Galton-Watson process. Adv. Appl. Prob. 1, 1–42.CrossRefGoogle Scholar
[2] Szekeres, G. (1958) Regular iteration of real and complex functions. Acta Math. 100, 203–258.CrossRefGoogle Scholar