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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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References

Agresti, A. (1975) On the extinction times of varying and random environment branching processes. J. Appl. Prob. 12, 39–46.CrossRefGoogle Scholar
Bellman, R. and Cooke, K. L. (1963) Differential-Difference Equations. Academic Press, New York.Google Scholar
Crump, K. S. and Mode, C. J. (1968) A general age-dependent branching process I. J. Math. Anal. Appl. 24, 494–508.CrossRefGoogle Scholar
Crump, K. S. and Mode, C. J. (1969) A general age-dependent branching process II. J. Math. Anal. Appl. 25, 8–17.CrossRefGoogle Scholar
Doney, R. A. (1972) A limit theorem for a class of supercritical branching processes. J. Appl. Prob. 9, 707–724.Google Scholar
Durham, S. D. (1971) Limit theorems for a general critical branching process. J. Appl. Prob. 8, 1–16.Google Scholar
Edler, L. (1976) Generationsabhängige Verzweigungsprozesse unter besonderer Berücksichtigung des Aussterbeverhaltens bei Bellman–Harris Prozessen mit exponentiellen Lebensdauern. Dissertation, Mainz.Google Scholar
Fearn, D. H. (1976) Supercritical age dependent branching processes with generation dependence. Ann. Prob. 4, 27–37.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications, Volume 2, 2nd edn. Wiley, New York.Google Scholar
Fildes, R. (1971) An age dependent branching process with variable lifetime distributions. Adv. Appl. Prob. 4, 453–474.Google Scholar
Fildes, R. (1974) An age dependent branching process with variable lifetime distributions: the generation size. Adv. Appl. Prob. 6, 291–308.CrossRefGoogle Scholar
Harris, T. E. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin.Google Scholar
Jagers, P. (1969) A general stochastic model for population development. Skand. Aktuarietidskr. 52, 84–103.Google Scholar
Jagers, P. (1975) Branching Processes with Biological Applications. Wiley, London.Google Scholar
Mode, C. J. (1974) Discrete time age-dependent branching processes in relation to stable population theory in demography. Math. Biosci. 19, 73–100.Google Scholar
Sewastjanow, B. A. (1975) Verzweigungsprozesse. Oldenbourg, München.CrossRefGoogle Scholar
Smith, W. L. (1954) Asymptotic renewal theorems. Proc. R. Soc. Edinburgh A 64, 9–48.Google Scholar
Smith, W. L. (1962) On some renewal theorems for nonidentically distributed variables. Proc. 4th Berkeley Symp. Math. Statist. Prob. 3, 467–514.Google Scholar
Smith, W. L. (1967) On the weak law of large numbers and the generalized elementary renewal theorem. Pacific J. Maths 22, 171–188.CrossRefGoogle Scholar