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November, 1985 Asymptotic Growth of Controlled Galton-Watson Processes
Petra Kuster
Ann. Probab. 13(4): 1157-1178 (November, 1985). DOI: 10.1214/aop/1176992802


The almost sure growth behavior of some time-homogeneous Markov chains is studied. They generalize the ordinary Galton-Watson processes with regard to allowing state-dependent offspring distributions and also to controlling the number of reproducing individuals by a random variable that depends on the state of the process. The main assumption is that the mean offspring per individual is nonincreasing while the state increases. These controlled Galton-Watson processes can be included in a general growth model whose divergence rate is determined. In case of processes that differ from the Galton-Watson process only by the state dependence of the offspring distributions, a necessary and sufficient moment condition for divergence with "natural" rate is obtained generalizing the $(x \log x)$ condition of Galton-Watson processes. In addition, some criteria are given when a state-dependent Galton-Watson process behaves like an ordinary supercritical Galton-Watson process.


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Petra Kuster. "Asymptotic Growth of Controlled Galton-Watson Processes." Ann. Probab. 13 (4) 1157 - 1178, November, 1985.


Published: November, 1985
First available in Project Euclid: 19 April 2007

zbMATH: 0576.60078
MathSciNet: MR806215
Digital Object Identifier: 10.1214/aop/1176992802

Primary: 60J80
Secondary: 60F15 , 60J10

Keywords: $\varphi$-controlled branching process , Galton-Watson process , Growth model , Growth rate , population-size-dependent branching process , state-dependent offspring distribution

Rights: Copyright © 1985 Institute of Mathematical Statistics

Vol.13 • No. 4 • November, 1985
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