Large deviations of branching process in a random environment. II

We consider the probabilities of large deviations for the branching process $ Z_n $ in a random environment, which is formed by independent identically distributed variables. It is assumed that the associated random walk $ S_n = \xi_1 + \ldots + \xi_n $ has a finite mean $ \mu $ and satisfies the Cramér condition $ E e^{h \xi_i} <\infty $, $ 0 <h <h^+$. Under additional moment constraints on $ Z_1 $, the exact asymptotic of the probabilities $ {\mathbf P} (\ln Z_n \in [x, x + \Delta_n)) $ is found for the values $ x/n $ varying in the range depending on the type of process, and for all sequences $ \Delta_n $ that tend to zero sufficiently slowly as $ n \to \infty $. A similar theorem is proved for a random process in a random environment with immigration.