Branching processes in random environment with cooling

It is well known that a branching process in random environment can be analyzed via the associated random walk $S_n = \xi_1 + \dotsb + \xi_n$, where $\xi_k = \ln \varphi_{\eta_k}'(1)$. Here $\varphi_x (t)$ and $\{ \eta_k \}_{k = 1}^{\infty}$ are the generating functions of the number of descendants of a paricle for given environment x and the random environment respectively. We study the probability of extinction of a branching process in random environment with cooling. In constract to classic BPRE, in this process every environment lasts for several generations. It turns out that this variant of BPRE is also closely related to a random walk $S_n = \tau_1 \xi_1 + \dotsb + \tau_n \xi_n$, where $\xi_k = \ln \varphi_{\eta_k}'(1)$. Here $\varphi_x (t)$ and $\{ \eta_k \}_{k = 1}^{\infty}$ are the generating functions of the number of descendants and the random environment respectively and $\tau_k$ is the duration of the $k$-th cooling. In this paper we find several sufficient conditions for extinction probability to be one or less than one correspondingly.