Large Deviations of a Strongly Subcritical Branching Process in a Random Environment

We consider probabilities of large deviations for a strongly subcritical branching process $\{Z_n,\, n\ge 0\}$ in a random environment generated by a sequence of independent identically distributed random variables. It is assumed that the increments of the associated random walk $S_n=\xi _1+\ldots +\xi _n$ have finite mean $\mu $ and satisfy the Cramér condition $\operatorname {\mathbf E}e^{h\xi _i}<\infty $, $0<h<h^+$. Under additional moment restrictions on $Z_1$, we find exact asymptotics of the probabilities $\operatorname {\mathbf P}(\ln Z_n \in [x,x+\Delta _n))$ with $x/n$ varying in the range $(0,\gamma )$, where $\gamma $ is a positive constant, for all sequences $\Delta _n$ that tend to zero sufficiently slowly as $n\to \infty $. This result complements an earlier theorem of the author on the asymptotics of such probabilities in the case where $x/n>\gamma $.