Population size of a critical branching process evolving in unfovarable environment

V. A. Vatutin ; E. E. Dyakonova

Teoriâ veroâtnostej i ee primeneniâ, Tome 68 (2023) no. 3, pp. 509-531

Résumé

Let $\{Z_n,\, n=0,1,\dots\}$ be a critical branching process in a random environment and let $\{S_n,\, n=0,1,\dots\}$ be its associated random walk. It is known that if the distribution of increments of this random walk belongs (without centering) to the domain of attraction of a stable distribution, then there is a sequence $a_1,a_2,\dots$ regularly varying at infinity such that, for any ${t\in (0,1]}$ and ${x\in(0,+\infty)}$, $\lim_{n\to \infty}\mathbf{P}({\ln Z_{nt}}/{a_n}\leq x\mid Z_n>0) = \lim_{n\to \infty}\mathbf{P}({S_{nt}}/{a_n}\leq x\mid {Z_n>0})=\mathbf{P}({Y_t^+\leq x})$, where $Y_{t}^{+}$ is the value at point $t$ of the meander of unit length of a strictly stable process. We complement this result with a description of conditional distributions of appropriately normalized random variables (r.v.'s) $\ln Z_{nt}$ and $S_{nt}$, given $\{S_n\leq\varphi(n);\ Z_n>0\}$, where $\varphi (n)\to \infty $ as $n\to \infty $ in such a way that $\varphi (n)=o(a_n)$.

Détail

Keywords: branching process, unfavorable random environment, survival probability.