Late Extinction of Markov Reccurence Sequence in a Random Environment

Let $\boldsymbol\eta=(\eta_1,\eta_2,\dotsc)$ be a sequence of i.i.d. r.v., called environment. Let $\{Y_n\}$ be a Markov chain on $\mathbb{N}\cup \{0\}$ such that

• for given $\boldsymbol\eta$ the sequence $\{Y_n\}$ is non-homogeneous Markov chain;
• we have a representation
  $$ Y_{n+1} = A_{n+1} Y_n + B_{n+1}, $$
  where $A_i = g_1(\eta_i)$ is a non-negative function of the environment and $(Y_0, B_1, B_2, \dotsc, B_i)$ is independent of $(A_{i+1},A_{i+2},\dotsc)$;
• for some $0<h^*<h$ the following neglibility condition on $\{B_i\}$ holds:
  \begin{equation} \label{Small} {\mathbf E}\left(\left.|B_{i+1}|^h\right|Y_i, \boldsymbol\eta\right)\le \zeta_{i+1} A_{i+1} Y_i^{h^*},\ i\ge 0. \end{equation}
  Here we make some moment assumptions on $\zeta_{i}=g_2(\eta_{i})$, $i\ge 0$.

We call $\{Y_n\}$ Markov linear reccurence sequence in a random environment (MRSRE). We assume that $\{0\}$ is an absorbing state and $\{Y_n\}$ on $\mathbb{N}$ is irreducible and aperiodic.
A natural example of MRSRE is a branching process in a random environment (BPRE) $Z_n$. In this case
$$ A_{n} = e^{\xi_{n}},\quad B_n = \sum_{i=1}^{Z_{n-1}} (X_{n,i} - e^{\xi_n}). $$
Here $X_{n,i}$ is a number of descendants of $i$-th particle in $(n-1)$-th generation, $\xi$ is a step of an associated random walk. Condition (\ref{Small}) is a corollary of Marcinkiewicz–Zygmund inequality.
Other examples of MRSRE are:

• bisexual branching processes in a random enviroment (BBPRE);
• maximal branching processes with c.d.f. $F(x) = 1- c/x + O(x^{-1-\delta})$, $x\to\infty$ for some positive $\delta$;
• maximal branching processes in a random enviroment with the conditional e.c.d.f. $F_{X|\eta}(x|y) = 1- c_y/x + r(x,y)/x^{1+\delta}$, $x\to\infty$, wherre $\delta$ is some positive number.

We also allow to add immigration to the processes above.
In the report we consider the limit behavior of
$$ {\mathbf P}(n<T<\infty),\ {\mathbf P}(Y_n=k|n<T<\infty) $$
as $n\to\infty$, where $T$ is the absorption time. We consider two cases, corresponding to supercritical (${\mathbf E}\ln A_1>0$) and subcritical (${\mathbf E}\ln A_1<0$) processes.
We show that in some cases (analogously to BPRE it's natural to call them strongly supercritical and strongly subcritical) it's true that
$$ {\mathbf P}(n<T<\infty)\sim C \rho^n,\quad {\mathbf P}(Y_n=k|n<T<\infty)\to p_k^*, $$
where $C$ is some positive constant, $\rho\in (0,1]$ – is some structural parameter of MRSRE, $\{p_k^*\}$ is some probability distribution. We can't describe $\rho$ explicitly (even for the case of supercritical BPRE), but obtain some limit relation on it.
In the report we'll discuss a general theory of $R$-positivity, apply the results to a general MRSRE and consider a particular case of BPRE, BBPRE, BPRE with immigration.