THE RATES OF GROWTH OF THE GALTON-WATSON PROCESS IN VARYING ENVIRONMENTS

Let {Z(n)} be a supercritical Galton-Watson process in varying environments, and W be the limit of the non-negative martingale {Z(n)/EZ(n)}. Under a condition which ensures that W is not identically equal to zero we give an upper bound on the possible rates of growth of the process on the set {W = 0}, and find a sufficient condition for the process to have only one rate of growth. We also give an example of a process whose offspring distributions have bounded pth moments, for some p > 1, and which has an infinite number of rates of growth.