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 ( Z n ) 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 P ( Z n ≥ e θ n ) is (on logarithmic scale) completely determined by a convex function Γ depending on properties of S . In many cases Γ is identical with the rate function of ( S n ) . However, if the branching process is strongly subcritical, there is a phase transition and the asymptotics of P ( Z n ≥ e θ n ) and P ( S n ≥ θ n ) differ for small θ .