On the prospective minimum of a random walk conditioned to stay nonnegative

Let
$$ S_{0}=0,\quad S_{n}=X_{1}+...+X_{n},\ n\geq 1, $$
be a random walk whose increments belong without centering to the domain of attraction of a stable law with scaling constants $a_{n}$, that provide convergence as $n\rightarrow \infty $ of the distributions of the sequence $ \left\{ S_{n}/a_{n},n=1,2,...\right\} $ to this stable law. Let $ L_{r,n}=\min_{r\leq m\leq n}S_{m}$ be the minimum of the random walk on the interval $[r,n]$. We show that
$$ \lim_{r,k,n\rightarrow \infty }\mathbf{P}\left( L_{r,n}\leq ya_{k}|S_{n}\leq ta_{k},L_{0,n}\geq 0\right) ,t\in \left( 0,\infty \right) $$
can have five different expressions, the forms of which depend on the relationships between the parameters $r,k$ and $n$.
The obtained results are used to study the distribution of the number of particles in a critical reduced branching process in a random environment.