On Conditioning a Random Walk to Stay Nonnegative

Abstract

Let $S$ be a real-valued random walk that does not drift to $\infty$, so $P(S_k \geq 0$ 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 $n \rightarrow \infty$. 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.