Geodesic
On large deviations of branching processes in a random environment: a geometric distribution of the number of descendants
Diskretnaya Matematika, Tome 18 (2006) no. 2, pp. 29-47
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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 $\boldsymbol{\mathsf P}(\ln Z_n>\theta n)$, $\theta>0$, 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. 
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M. V. Kozlov. On large deviations of branching processes in a random environment: a geometric distribution of the number of descendants. Diskretnaya Matematika, Tome 18 (2006) no. 2, pp. 29-47. http://geodesic.mathdoc.fr/item/DM_2006_18_2_a1/

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