Asymptotics of surviving probability of critical branching processes in random environment with cooling

It is well known that branching processes in random environment (BPRE) can be described in terms of the associated random walk

$$ S_n = \xi_1 + \ldots + \xi_n, $$

where $\xi_k = \ln \varphi_{\eta_k}'(1)$, $\varphi_x (t)$ and $\eta_k$ are probability generating function and random environment respectively. The talk will address the issue of extinction of a branching process in random environment with cooling with $\mathsf{E} \xi_1 = 0$, where, in contrast to the classical BPRE, the $n$th environment lasts for $\tau_n$ generations. It turns out that this modification of BPRE is also closely related to the random walk

$$ S_n = \tau_1 \xi_1 + \ldots + \tau_n \xi_n, $$

where $\xi_k = \ln \varphi_{\eta_k}'(1)$, $\varphi_x (t)$ and $\eta_k$ are probability generating functions and random environment respectively and $\tau_k$ is called the $k$th cooling.
In this talk we will show that if the number of offsprings of any particle has geometric distribution and if the moments of $\xi$ and $\{ \tau_n \}_{n = 1}^{\infty}$ satisfy some assumptions, then the survival probability of the process satisfies the following asymptotic relation

$$ \mathsf{P} \left( Z_{s_n} > 0 \right) \sim \frac{c}{\sqrt{\tau_1^2 + \ldots + \tau_n^2}},~n \to \infty $$

for some positive $c$, where $s_n = \tau_1 + \ldots + \tau_n$.