Branching processes in random environment with freezing

It is well known that a branching process in random environment (BPRE) can be analyzed via the associated random walk
\begin{equation*}S_n = \xi_1 + \dotsb + \xi_n,\end{equation*}
where $\xi_k = \ln \varphi_{\eta_k}'(1)$. Here $\{ \eta_k \}_{k = 1}^{\infty}$ is the random environment and $\varphi_x (t)$ is the generating function of the number of descendants of a particle for given environment $x$. We study the probability of extinction of a branching process in random environment with freezing: in constrast to classic BPRE, in this process every state $\eta_k$ of the environment lasts for given number $\tau_k$ of generations. It turns out that this variant of BPRE is also closely related to a random walk
\begin{equation*}S_n = \tau_1 \xi_1 + \dotsb + \tau_n \xi_n.\end{equation*}
We find several sufficient conditions for extinction probability of such process to be one or less than one correspondingly.