On large deviations of branching processes in a random environment: geometric distribution 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 SmithWilkinson model) is considered. The asymptotics of large deviation probabilities P(ln Z(n) > theta n), theta > 0, are found provided that the steps of the accompanying random walk S-n satisfy the Cramer condition. In the cases of supercritical, critical, moderate, and intermediate subcritical processes the asymptotics follow that of the large deviations probabilities P(S-n <= theta n). In strongly subcritical case the same asymptotics hold for theta greater than some theta* (for theta <= theta* the asymptotics of large deviation probabilities are different).