Asymptotic Growth of Controlled Galton-Watson Processes

Kuster, Petra

Ann. Probab., Tome 13 (1985) no. 4, p. 1157-1178 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 1985-11-14
Classification:  Galton-Watson process,  state-dependent offspring distribution,  $\varphi$-controlled branching process,  population-size-dependent branching process,  growth model,  growth rate,  60J80,  60F15,  60J10