Some functionals for random walks and critical branching processes in extreme random environment

Abstract

Let $\left\{S_n,n\ge 0\right\}$ be a random walk whose increment distribution belongs without centering to the domain of attraction of an -stable law, i.e., there are some scaling constants $a_n$ such that the sequence $S_n/a_n,n=1,2,...,$ weakly converges, as to a random variable having an $\alpha$-stable distribution. Let $S_0=0,$% \begin{equation*} L_{n}:=\min \left( S_{1},...,S_{n}\right) ,τ_{n}:=\min \left\{ 0\leq k\leq n:S_{k}=\min (0,L_{n})\right\} . \end{equation*}% Assuming that $S_n\le h(n),$ where $h(n)$ is $o(a_n)$ and exists we prove several limit theorems describing the asymptotic behavior of the functionals \begin{equation*} \mathbf{E}\left[ e^{S_{τ_{n}}};S_{n}\leq h(n)\right] \end{equation*}% as $n\to \infty$. The obtained results are applied for studying the survival probability of a critical branching process evolving in an extremely unfavorable random environment. Key words: random walk, branching processes, random environment, survival probability, unfavorable environment

28 pages