Let S be a real-valued random walk that does not drift to ∞, so P(Sₖ ≥ 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 → ∞. 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.
No takes yet. Share an insight, caveat, or question.
Bertoin et al. (1994) studied this question.