On large deviations of branching processes in a random environment: a geometric distribution of the number of descendants

A branching process $Z_n$ with geometric distribution of descendants in a random environment represented by a sequence of independent identically distributed random variables (the Smith–Wilkinson model) is considered. The asymptotics of large deviation probabilities $\boldsymbol{\mathsf P}(\ln Z_n>\theta n)$, $\theta>0$, are found provided that the steps of the accompanying random walk $S_n$ satisfy the Cramér condition. In the cases of supercritical, critical, moderate, and intermediate subcritical processes the asymptotics follow that of the large deviations probabilities $\boldsymbol{\mathsf P}(S_n\le\theta n)$. In strongly subcritical case the same asymptotics hold for $\theta$ greater than some $\theta^*$ (for $\theta\le\theta^*$ the asymptotics of large deviation probabilities are different).
This research was supported by the Russian Foundation for Basic Research, grant 04–01–00700, and by DFG, project 436 RUS 113/722.