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A generalization of Goldstein's comparison lemma and the exponential limit law in critical Crump-Mode-Jagers branching processes

Published online by Cambridge University Press:  01 July 2016

John M. Holte
John M. Holte*
Affiliation:
Rensselaer Polytechnic Institute, Troy, New York
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Abstract

Let Z(t) be the population size at time t in a general age-dependent branching process (as defined by Crump and Mode, or Jagers) in which the number N of offspring of a parent has expected value 1 (critical case). Assuming positivity and finiteness of the second moments of N, of the lifespan distribution and of the expected number of births per parent as a function of age (also assumed to be strongly non-lattice), the distribution of Z(t)/t conditioned on non-extinction at time t is asymptotically exponential. The main step in the proof is a comparison lemma for the probability generating functions of Z(t) and of the embedded generation process.

Keywords

BRANCHING PROCESSESCRITICALCONDITIONED LIMIT LAWEMBEDDED GALTON–WATSON PROCESS
Type
Research Article
Information
Advances in Applied Probability , Volume 8 , Issue 1 , March 1976 , pp. 88 - 104
Copyright
Copyright © Applied Probability Trust 1976 

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References

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