General branching processes in varying environment (Q1327611)

In a general branching process each individual is equipped with an independent reproduction process giving her age at the birth of each of her children and each child's type. The distribution of this reproduction process depends on the individual' type. Here this distribution also depends on the environment, or to put this more simply, on the actual time of an individual's birth. This is, of course, a very general model, including many special cases that have already been studied. The paper shows, through conditioning and martingale methods, that certain properties of the expectation of the process, and a uniform integrability assumption, suffice for the convergence of the process itself. Conditions are also given for the limits arising to be continuous.