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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   P (ln  Z n   &gt;  θn ),  θ  &gt; 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   P ( S n   ≤  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).

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