Random walks conditioned to stay non-negative and branching processes in non-favorable random environment

Let $\{S_n,n\ge 0\}$ be a random walk whose increments belong without centering to the domain of attraction of an $\alpha$-stable law $\{Y_t,t\ge 0\}$, i.e. $S_{nt}\mathrm{/}{a}_n\Rightarrow Y_t,t\ge 0,$ for some scaling constants $a_n$. Assuming that $S_0=o(a_n)$ and $S_n\le \phi (n)=o(a_n),$ we prove several conditional limit theorems for the distribution of $S_{n-m}$ given $m=o(n)$ and ${\min }_{0\le k\le n}{S}_k\ge 0$. These theorems complement the statements established by F. Caravenna and L. Chaumont in 2013. The obtained results are applied for studying the population size of a critical branching process evolving in non-favorable environment.