Large deviations of branching process in a random environment

In this first part of the paper we find the asymptotic formulas for the probabilities of large deviations of the sequence defined by the random difference equation $Y_{n+1}=A_{n} Y_n + B_n$, where $A_1,A_2,\ldots$ are independent identically distributed random variables and $B_n$ may depend on $\{(A_k,B_k),0\leqslant k<n\}$ for any $n\geqslant1$. In the second part of the paper this results are applied to the large deviations of branching processes in a random environment.