Some functionals for random walks and critical branching processes in an extremely unfavourable random environment

V. A. Vatutin   ; C. Dong   ; E. E. Dyakonova

Sbornik. Mathematics, Tome 215 (2024) no. 10, pp. 1321-1350

Résumé

Let $\{S_{n},\,n\geq 0\}$ be a random walk whose increment distribution belongs without centering to the domain of attraction of an $\alpha$-stable law, that is, there are scaling constants $a_{n}$ such that the sequence $S_{n}/a_{n}$, $n=1,2,\dots$, converges weakly, as $n\to\infty$, to a random variable having an $\alpha$-stable distribution. Let $S_{0}=0$, $$ L_{n}:=\min (S_{1},\dots,S_{n})\quad\text{and}\quad\tau_{n}:=\min \{ 0\leq k\leq n\colon S_{k}=\min (0,L_{n})\}. $$ Assuming that $S_{n}\leq h(n)$, where $h(n)$ is $o(a_{n})$ as $n\to\infty$ and the limit $\lim_{n\to\infty}h(n)\in [-\infty,+\infty]$ exists, we prove several limit theorems describing the asymptotic behaviour of the functionals $$ \mathbf{E}[ e^{\lambda S_{\tau_{n}}};\, S_{n}\leq h(n)], \qquad \lambda>0, $$ as $n\to\infty$. The results obtained are applied to study the survival probability of a critical branching process evolving in an extremely unfavourable random environment. Bibliography: 15 titles.

Détail

Keywords: stable random walks, branching processes, survival probability, extreme random environment.

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