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)(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 SS, the asymptotics of P(Zn≥eθn) is (on logarithmic scale) completely determined by a convex function ΓΓ depending on properties of SS. In many cases ΓΓ is identical with the rate function of (Sn)(Sn). However, if the branching process is strongly subcritical, there is a phase transition and the asymptotics of P(Zn≥eθn) and P(Sn≥θn) differ for small θθ.