Renewal theorems and stability for the reflected process

Renewal-like results and stability theorems relating to the large-time behaviour of a random walk Sn reflected in its maximum, Rn = max0 ≤ j ≤ n Sj - Sn, are proved. Mainly, we consider the behaviour of the exit time, τ (r), where τ (r) = min {n ≥ 1 : Rn > r}, r > 0, and the exit position, Rτ (r), as r grows large, with particular reference to the cases when Sn has finite variance, and/or finite mean. Thus, limr → ∞ E Rτ (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 Rτ (r) / r over(→, P) 1 as r → ∞ and limr → ∞ Rτ (r) / r = 1 almost surely (a.s.); alternatively expressed, the overshoot Rτ (r) - r is o (r) as r → ∞, in probability or a.s. Comparisons are also made with exit times of the random walk Sn across both two-sided and one-sided horizontal boundaries. © 2008 Elsevier B.V. All rights reserved.