Large deviations for branching processes in random environment

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 at most geometric tails, we derive an upper bound for the tail probabilities. Under mild regularity of the associated random walk $S$, the asymptotics of $\mathsf{P}(Z_n\geq e^{\theta n})$ is—on a logarithmic scale—completely determined by a convex function $\Gamma$ depending on properties of the associated random walk. In many cases $\Gamma$ is identical with the rate function of $(S_n)$. However, if the associated random walk has a strong negative drift, the asymptotics of $\mathsf{P}(Z_n\geq e^{\theta n})$ and $\mathsf{P}(S_n\geq \theta n)$ are different for small $\theta$.