Abstract:
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.