Let $\{ S_n,n\geq 0\} $ be a random walk whose increments belong without centering to the domain of attraction of an $\alpha$-stable law $\{ Y_t,t\geq 0\}$, i.e. $S_{ nt}/a_n\Rightarrow Y_t,t\geq 0,$ for some scaling constants $a_n$. Assuming that $S_0=o(a_{ n})$ and $S_n\leq \varphi (n)=o(a_n),$ we prove several conditional limit theorems for the distribution of $S_{ n-m}$ given $m=o(n)$ and $\min_{ 0\leq k\leq n}S_k\geq 0$. 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.
Categories: math.PR