On Conditioning a Random Walk to Stay Nonnegative

Abstract

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