Random walks conditioned to stay nonnegative and branching processes in an unfavourable environment

Let $\{S_n,\,n\geqslant 0\}$ be a random walk with increments that belong (without centering) to the domain of attraction of an $alpha$-stable law, that is, there exists a process $\{Y_t,\,t\geqslant 0\}$ such that $S_{nt}/a_{n}$ $\Rightarrow$ $Y_t$, $t\geqslant 0$, as $n\to\infty$ for some scaling constants $a_n$. Assuming that $S_{0}=o(a_n)$ and $S_n\leqslant \varphi (n)=o(a_n)$, we prove several conditional limit theorems for the distribution of the random variable $S_{n-m}$ given that $m=o(n)$ and $\min_{0\leqslant k\leqslant n}S_k\geqslant 0$. These theorems supplement the assertions established by Caravenna and Chaumont in 2013. Our results are used to study the population size of a critical branching process evolving in an unfavourable environment.
Bibliography: 28 titles.