On conditioning a random walk to stay nonnegative

Let S be a real-valued random walk that does not drift to ∞, so P(S k ≥ 0 for all k)=0. We condition S to exceed n before hitting the negative halfline, respectively, to stay nonnegative up to time n. We study, under various hypotheses, the convergence ofthese conditional laws as n →∞. First, when S oscillates, the two approximations converge to the same probability law. This feature may be lost when S drifts to -∞. Specifically, under suitable assumptions on the upper tail of the step distribution, the two approximations then converge to different probability laws.