Renewal Theorems and Stability for the Reflected Process

Renewal-like results and stability theorems relating to the largetime 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→∞ERτ(r)/r = 1 is shown to hold when E|X| < ∞ and EX < 0 or EX2 < ∞ and EX = 0, and in these situations Eτ(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 P → 1 as r → ∞ and limr→∞Rτ(r)/r = 1 almost surely (a.s.). Comparisons are also made with exit times of the random walk Sn across both two-sided and one-sided horizontal boundaries. 2000 MSC Subject Classifications: primary: 60J15, 60F15, 60K05, 60G40 secondary: 60F05, 60G42