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

Let S-0=0, S-n = X-1 + ... + X-n,X- n >= 1, be a random walk whose increments belong without centering to the domain of attraction of a stable law with scaling constants an, that provide convergence as n ->infinity of the distributions of the elements of the sequence {S-n/a(n),n=1,2,...} to this stable law. Let L-r,L-n = min(r <= m <= n)S(m) be the minimum of the random walk on the interval [r,n]. It is shown that lim(r,k,n ->infinity)P(L-r,L-n <= ya(k )| S-n <= ta(k), L-0,L-n >= 0),t is an element of (0,infinity), can have five different expressions, the forms of which depend on the relationships between the parameters r,k and n.