A UNIFYING APPROACH TO BRANCHING PROCESSES IN A VARYING ENVIRONMENT

Branching processes (Z_n)_{n \ge 0} in a varying environment generalize the Galton-Watson process, in that they allow time dependence of the offspring distribution. Our main results concern general criteria for almost sure extinction, square integrability of the martingale (Z_n/\mathrm E[Z_n])_{n \ge 0}, properties of the martingale limit W and a Yaglom-type result stating convergence to an exponential limit distribution of the suitably normalized population size Z_n, conditioned on the event Z_n \gt 0. The theorems generalize/unify diverse results from the literature and lead to a classification of the processes.