Stability and other limit laws for exit times of random walks from a strip or a halfplane

We show that the passage time, T*(r), of a random walk Sn above a horizontal boundary at r (r ≥ 0) is stable (in probability) in the sense that T*(r)/C(r) →P 1 as r → ∞ for a deterministic function C(r) > 0, if and only if the random walk is relatively stable in the sense that S_n/B_n →P 1 as n → ∞ for a deterministic sequence B_n > 0. The stability of a passage time is an important ingredient in some proofs in sequential analysis, where it arises during applications of Anscombe's Theorem. We also prove a counterpart for the almost sure stability of T*(r), which we show is equivalent to E|X| < ∞, EX > 0. Similarly, counterparts for the exit of the random walk from the strip {|y| ≤ r} are proved. The conditions are further related to the relative stability of the maximal sum and the maximum modulus of the sums. Another result shows that the exit position of the random walk outside the boundaries at ±r drifts to ∞ as r → ∞ if and only if the random walk drifts to ∞.