Let \begin{ equation*} S_{ 0}=0,\quad S_{ n}=X_{ 1}+...+X_{ n},\ n\geq 1, \end{ equation*} 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 elements 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 $[r,n]$. It is shown that \begin{ equation*} \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), \end{ equation*} can have five different expressions, the forms of which depend on the relationships between the parameters $r,k$ and $n$.
Categories: math.PR