An inhomogeneous controlled branching process∗

Abstract

We consider a discrete-time branching process in which the offspring distribution is generation-dependent and the number of reproductive individuals is controlled by a random mechanism. This model is a Markov chain, but, in general, the transition probabilities are nonstationary. Under not too restrictive hypotheses, this model presents the classical duality of branching processes: it either becomes extinct or grows to infinity. Sufficient conditions for the almost sure extinction and for a positive probability of indefinite growth are given. Finally, the rates of growth of the process are studied, provided that there is no extinction.