Renewal theorems and stability for the reflected process

Renewal-like results and stability theorems relating to the large-time behaviour of a random walk S_n reflected in its maximum, R_n = max_{0 ≤ j ≤ n} S_j - S_n, are proved. Mainly, we consider the behaviour of the exit time, τ (r), where τ (r) = min {n ≥ 1 : R_n > r}, r > 0, and the exit position, R_{τ (r)}, as r grows large, with particular reference to the cases when S_n has finite variance, and/or finite mean. Thus, lim_{r → ∞} E R_{τ (r)} / r = 1 is shown to hold when E | X | < ∞ and E X < 0 or E X² < ∞ and E X = 0, and in these situations E τ (r) grows like a multiple of r, or of r², respectively. More generally, under only a rather mild side condition, we give equivalences for R_{τ (r)} / r over(→, P) 1 as r → ∞ and lim_{r → ∞} R_{τ (r)} / r = 1 almost surely (a.s.); alternatively expressed, the overshoot R_{τ (r)} - r is o (r) as r → ∞, in probability or a.s. Comparisons are also made with exit times of the random walk S_n across both two-sided and one-sided horizontal boundaries.