A branching process Zn 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 P(ln Zn > θn), θ > 0, are found provided that the steps of the accompanying random walk Sn satisfy the Cramér condition. In the cases of supercritical, critical, moderate, and intermediate subcritical processes the asymptotics follow that of the large deviations probabilities P(Sn ≤ θn). In strongly subcritical case the same asymptotics hold for θ greater than some θ* (for θ ≤ θ* the asymptotics of large deviation probabilities are different).
Copyright 2006, Walter de Gruyter
