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Seminar by Department of Discrete Mathematic, Steklov Mathematical Institute of RAS
October 8, 2024 16:00, Moscow, Steklov Mathematical Institute, Room 313 (8 Gubkina) + online


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

I. D. Korshunovab

a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
b Lomonosov Moscow State University, Faculty of Mechanics and Mathematics



Abstract: 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 $s_n = \tau_1 + \ldots + \tau_n$.


© Steklov Math. Inst. of RAS, 2024