Renewal theorems and stability for the reflected process

Renewal-like results and stability theorems relating to the large-time behaviour of a random walk SnSn reflected in its maximum, Rn=max0≤j≤nSj−SnRn=max0≤j≤nSj−Sn, are proved. Mainly, we consider the behaviour of the exit time, τ(r)τ(r), where τ(r)=min{n≥1:Rn>r},r>0, and the exit position, Rτ(r)Rτ(r), as rr grows large, with particular reference to the cases when SnSn has finite variance, and/or finite mean. Thus, limr→∞ERτ(r)/r=1limr→∞ERτ(r)/r=1 is shown to hold when E|X|<∞E|X|<∞ and EX<0EX<0 or EX2<∞EX2<∞ and EX=0EX=0, and in these situations Eτ(r)Eτ(r) grows like a multiple of rr, or of r2r2, respectively. More generally, under only a rather mild side condition, we give equivalences for Rτ(r)/r→P1 as r→∞r→∞ and limr→∞Rτ(r)/r=1limr→∞Rτ(r)/r=1 almost surely (a.s.); alternatively expressed, the overshoot Rτ(r)−rRτ(r)−r is o(r)o(r) as r→∞r→∞, in probability or a.s. Comparisons are also made with exit times of the random walk SnSn across both two-sided and one-sided horizontal boundaries.