On positiveness of markov chain in a random environment with absorbing at zero

In the report I will tell about joint results with P. Senko, generalizing classical theorem of Afanasyev, Geiger, Kersting, Vatutin for branching processes in a random environment (BPRE).
Consider the sequence
$$ Y_{n+1} = A_n Y_n + B_n. $$
Here $\{A_n\}$ are i.i.d, $B_n$ are possible dependent (but independent of $(A_{n+1},A_{n+2},\dotsc)$) and non-identically distributed, but satisfying the equation
$$ {\mathbf E}(\left|B_{n+1}\right|^{1+\delta}|Y_n)\le d(\eta_{n+1}) Y_n $$
for some $\delta>0$ and some measurable function $d$.
Moreover, assume that $\{Y_n\}$ is an integer-valued Markov chain in a random environment $\boldsymbol\eta=(\eta_1,\eta_2,\dotsc)$, where $\eta_i$ are i.i.d. random variables. We assume that $\{0\}$ is an absorbing state.
This model inculdes a lot of branching processes, particularly, BPRE and bisexual BPRE.
The main result of the work describes the asymptotics of
$$ {\mathbf P}(Y_n>0) $$
as $n\to\infty$. The goal of report is to describe the main steps of the proof, especially how to avoid the usage of p.g.f.'s.