On large deviations of a branching process in random environments with immigration at moments of extinction

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 $P(\text{ln}\, Z_{n}^{*}\geqslant\theta n)$.