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 X 2 < ∞ and E X = 0 , and in these situations E τ ( r ) grows like a multiple of r , or of r 2 , respectively. More generally, under only a rather mild side condition, we give equivalences for R τ ( r ) / r → 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.