Geodesic
On large deviations of a branching process in random environments with immigration at moments of extinction
Diskretnaya Matematika, Tome 26 (2014) no. 4, pp. 36-42.

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Let ($Z_{n}^{*}$) be a branching process in independent identically distributed random environments with (conditioned on the environments) geometric distribution of the number of offsprings and immigration of independent identically distributed numbers of new particles at moments of extinction. Supposing that the increments of the accompanying random walk ($S_n$) and numbers of immigrants satisfy right-hand Cramér condition we obtain the asymptotics of large deviation probabilities $P(\text{ln}\, Z_{n}^{*}\geqslant\theta n)$.
Keywords: large deviations, Cramér condition, branching processes, random environments, processes with immigration. 
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D. V. Dmitrushchenkov. On large deviations of a branching process in random environments with immigration at moments of extinction. Diskretnaya Matematika, Tome 26 (2014) no. 4, pp. 36-42. http://geodesic.mathdoc.fr/item/DM_2014_26_4_a3/

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