Curve crossing for random walks reflected at their maximum

Let R_n = max_0≤j≤n S_j - S_n be a random walk S_n reflected in its maximum. Except in the trivial case when P(X ≥ 0) = 1, R_n will pass over a horizontal boundary of any height in a finite time, with probability 1. We extend this by giving necessary and sufficient conditions for finiteness of passage times of R _n above certain curved (power law) boundaries, as well. The intuition that a degree of heaviness of the negative tail of the distribution of the increments of S_n is necessary for passage of R_n above a high level is correct in most, but not all, cases, as we show. Conditions are also given for the finiteness of the expected passage time of R_n above linear and square root boundaries.