On the prospective minimum of the random walk conditioned to stay non-negative

Let {equation*} S₀=0, Sₙ=X₁+...+Xₙ,\ n≥ 1, {equation*} 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→ ∞ of the distributions of the elements of the sequence \ Sₙ/aₙ,n=1,2,...\ to this stable law. Let Lr,n=minr≤ m≤ nSₘ be the minimum of the random walk on the interval $[r,n]$. It is shown that {equation*} limr,k,n→ ∞P( Lr,n≤ yaₖ|Sₙ≤ taₖ,L0,n≥ 0) ,\, t∈ ( 0,∞ ), {equation*} can have five different expressions, the forms of which depend on the relationships between the parameters $r,k$ and n.