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Some functionals for random walks and critical branching processes in an extremely unfavourable random environment
V. A. Vatutina,
C. Dongb,
E. E. Dyakonovaa a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
b Xidian University, Xi'an, P. R. China
Abstract:
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.
Keywords:
stable random walks, branching processes, survival probability, extreme random environment.
MSC: Primary
60G50; Secondary
60J80,
60K37 Received: 13.02.2024 and 01.07.2024
DOI:
10.4213/sm10081
Fulltext:
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