The Annals of Probability
- Ann. Probab.
- Volume 13, Number 4 (1985), 1157-1178.
Asymptotic Growth of Controlled Galton-Watson Processes
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.
Ann. Probab., Volume 13, Number 4 (1985), 1157-1178.
First available in Project Euclid: 19 April 2007
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Kuster, Petra. Asymptotic Growth of Controlled Galton-Watson Processes. Ann. Probab. 13 (1985), no. 4, 1157--1178. doi:10.1214/aop/1176992802. https://projecteuclid.org/euclid.aop/1176992802