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Geometric rate of growth in population-size-dependent branching processes

Published online by Cambridge University Press:  14 July 2016

F. C. Klebaner
F. C. Klebaner*
Affiliation:
University of Melbourne
*
∗ Postal address: Department of Statistics, Richard Berry Building, University of Melbourne, Parkville, VIC 3052, Australia.
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Abstract

We consider a branching-process model {Zn}, where the law of offspring distribution depends on the population size. We consider the case when the means mn (mn is the mean of offspring distribution when the population size is equal to n) tend to a limit m > 1 as n →∞. For a certain class of processes {Zn} necessary conditions for convergence in L1 and L2 and sufficient conditions for almost sure convergence and convergence in L2 of Wn = Zn/mn are given.

Keywords

STOCHASTIC POPULATION MODELSHOMOGENEOUS MARKOV CHAINSMARTIN-GALESLIMIT THEOREMS
Type
Research Papers
Information
Journal of Applied Probability , Volume 21 , Issue 1 , March 1984 , pp. 40 - 49
Copyright
Copyright © Applied Probability Trust 1984 

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References

References

[1] Fujimagari, T. (1976) Controlled Galton–Watson process and its asymptotic behaviour. Kodai Math. Sem. Rep. 27, 11–18.Google Scholar
[2] Klebaner, F. C. (1983) Population-size-dependent branching process with linear rate of growth. J. Appl. Prob. 20, 242–250.Google Scholar
[3] Klebaner, F. C. (1984) On population-size-dependent branching processes. Adv. Appl. Prob. 16.Google Scholar
[4] Knopp, K. (1928) Theory and Applications of Infinite Series. Blackie & Sons, London.Google Scholar
[5] Labrovskii, V. A. (1972) A limit theorem for generalized branching process depending on the size of the population. Theory Prob. Appl. 17, 72–85.Google Scholar
[6] Levina, L. V., Leontovich, A. M. and Piateski–Shapiro, I. I. (1968) On regulative branching process. Problemy Pederaci Informacii 4, 72–82.Google Scholar
[7] Loève, M. (1978) Probability Theory II. Springer-Verlag, New York.Google Scholar
[8] Zubkov, A. M. (1974) Analogies between Galton–Watson processes and f branching processes. Theory Prob. Appl. 19, 309–331.Google Scholar

Reference added in proof

[9] Höpfner, R. (1983) On some classes of population-size-dependent Galton–Watson processes. Submitted to J. Appl. Prob. Google Scholar