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Strict supercritical generation-dependent Crump–Mode–Jagers branching processes

Published online by Cambridge University Press:  01 July 2016

L. Edler
L. Edler*
Affiliation:
University of Mainz
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Abstract

The general age-dependent branching model of Crump, Mode and Jagers will be generalized towards generation-dependent varying lifespan and reproduction distributions. A system of integral and renewal equations is established for the generating functions and the first two moments of Zi(t) (the number of individuals alive at time t), if the population was initiated at time 0 by one ancestor of age 0 from generation i. Convergence in quadratic mean of Zi(t)/EZi(t) as t tends to infinity is obtained if the generation-dependent reproduction functions converge to a supercritical one. In particular, if this convergence is slow enough tγ exp (αt) is the asymptotic behavior of EZi(t) for t tending to infinity, where γ is a positive real number and α the Malthusian parameter of growth of the limiting reproduction function.

Keywords

BRANCHING PROCESSCRUMP-MODE-JAGERS PROCESSBELLMAN-HARRIS PROCESSGENERATION-DEPENDENTSUPERCRITICALREPRODUCTION FUNCTIONL2-CONVERGENCERENEWAL THEORY
Type
Research Article
Information
Advances in Applied Probability , Volume 10 , Issue 4 , December 1978 , pp. 744 - 763
Copyright
Copyright © Applied Probability Trust 1978 

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