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The Galton-Watson process conditioned on the total progeny

Published online by Cambridge University Press:  14 July 2016

Douglas P. Kennedy
Douglas P. Kennedy*
Affiliation:
University of Sheffieldcor1corresp
*
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Abstract

Let Zk denote the number in the kth generation of a Galton-Watson process initiated by one individual and let N be the total progeny, i.e., As n → ∞ the limiting behaviour of the process {Zk, 0 ≦ k ≦ n} conditioned on the event {N =n} is studied. The results obtained are of exactly the same form for the subcritical, critical and supercritical cases. This is in marked contrast to the analogous situation got by conditioning on non-extinction by the nth generation and letting n → ∞. In the latter case the limiting results differ in form for the critical and non-critical cases.

Keywords

BRANCHING PROCESSESGALTON-WATSON PROCESSTOTAL PROGENYLIMIT THEOREMSCONDITIONED LIMIT THEOREMS
Type
Short Communications
Information
Journal of Applied Probability , Volume 12 , Issue 4 , December 1975 , pp. 800 - 806
Copyright
Copyright © Applied Probability Trust 1975 

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References

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