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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

Information

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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