Critical branching processes evolving in a unfavorable random environment

V. A. Vatutin ; E. E. Dyakonova

Diskretnaya Matematika, Tome 34 (2022) no. 3, pp. 20-33

Résumé

Let $\{Z_{n},n=0,1,2,\dots\}$ be a critical branching process in a random environment, and $\{S_{n},n=0,1,2,\dots\}$ be its associated random walk. It is known that if the increments of this random walk belong (without centering) to the domain of attraction of a stable law, then there exists a regularly varying at infinity sequence $a_{1},a_{2},\dots$ such that conditional distributions \begin{equation*} \mathbf{P}\bigg(\frac{S_{n}}{a_{n}}\leq x\Bigm| Z_{n}>0\bigg),\quad x\in(-\infty,+\infty), \end{equation*} converge weakly to the distribution of strictly positive proper random variable. In this paper we add to this result the description of the asymptotic behavior of the probability \begin{equation*} \mathbf{P}(Z_{n}>0, S_{n}\leq \varphi(n)), \end{equation*} where $\varphi (n)\to \infty$ for $n\to \infty$ in such a way that $\varphi (n)=o(a_{n})$.

Détail

Keywords: branching process, unfavorable random environment, non-extinction probability.