Final product of a random recurrence sequence

Consider a model of a random recurrence sequence. Let $(A_i, B_i)$, $i\ge0$, be two-dimensional random vectors. Then $Y_n = A_{n-1} Y_{n-1} + B_{n-1}$, $n \in \mathbb{N}$, is called a random recurrence sequence. The model was introduced and studied by A.V. Shklyaev.
It is quite interesting to study such sequences because a lot of models of branching processes can be represented as random recurrence sequences. For instance, branching process in a random environment with and without immigration, bisexual branching process in a random environment and many others.
The next step in the study of random recurrence sequences is to examine their final product. Suppose $Y_n$ is the random recurrence sequence with positive integer values. Let $C_{n,i}$, $n\in \mathbb{N}$, $i\ge0$, be independent identically distributed random variables. The random variable
\begin{eqnarray} L_{n}=\sum_{i=1}^{Y_n} C_{n,i} \notag \end{eqnarray}
is called the local final product of the random recurrence sequence $Y_n$. The overall final product $F_n$ is defined as
\begin{eqnarray} F_{n}=\sum_{m=1}^{n} \sum_{i=1}^{Y_m} C_{m,i}. \notag \end{eqnarray}

In the report author will show a theorem about large deviations probabilities for $L_n$ and $F_n$. To be more specific, we study the probabilities ${\mathbf P}\left(\ln{L_n}\in\left[x,x+\Delta_n\right)\right)$ and ${\mathbf P}\left(\ln{F_n}\in\left[x,x+\Delta_n\right)\right)$ as $n\to\infty$, where $n^{-1}x$ varies in certain range and $\Delta_n$ tends to zero slowly enough.
The applications of this result to branching processes in a random environment also will be presented in the report.