Upper large deviations of branching processes in a random environment--Offspring distributions with geometrically bounded tails

We generalize a result by Kozlov on large deviations of branching processes (Zn) in an i.i.d. random environment. Under the assumption that the offspring distributions have geometrically bounded tails and mild regularity of the associated random walk S, the asymptotics of is (on logarithmic scale) completely determined by a convex function [Gamma] depending on properties of S. In many cases [Gamma] is identical with the rate function of (Sn). However, if the branching process is strongly subcritical, there is a phase transition and the asymptotics of and differ for small [theta].