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

Let $\{S_n,n\ge 0\}$ be a random walk whose increment distribution belongs without centering to the domain of attraction of an $$-stable law, i.e., there are some scaling constants $a_n$ such that the sequence $S_n\mathrm{/}{a}_n,n=1,2,\mathrm{.}\mathrm{.}\mathrm{.},$ weakly converges, as $$ 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 $n\to \mathrm{\infty }$. 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