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Growth of Integral Transforms and Extinction in Critical Galton-Watson Processes

Part of: Markov processes Integral equations, integral transforms Limit theorems

Published online by Cambridge University Press:  14 July 2016

Daniel Tokarev
Daniel Tokarev*
Affiliation:
The University of Melbourne
*
∗Postal address: Australian Research Council Centre of Excellence for Mathematics and Statistics of Complex Systems (MASCOS), The University of Melbourne, 139 Barry Street, Carlton, VIC 3010, Australia. Email address: dtokarev@unimelb.edu.au
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Abstract

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The mean time to extinction of a critical Galton-Watson process with initial population size k is shown to be asymptotically equivalent to two integral transforms: one involving the kth iterate of the probability generating function and one involving the generating function itself. Relating the growth of these transforms to the regular variation of their arguments, immediately connects statements involving the regular variation of the probability generating function, its iterates at 0, the quasistationary measures, their partial sums, and the limiting distribution of the time to extinction. In the critical case of finite variance we also give the growth of the mean time to extinction, conditioned on extinction occurring by time n.

Keywords

Critical Galton-Watson processbranching processintegral transformregular variationmean time to extinctionTauberian theorem

MSC classification

Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F05: Central limit and other weak theorems 65R10: Integral transforms
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 2 , June 2008 , pp. 472 - 480
Copyright
Copyright © Applied Probability Trust 2008 

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