Let Rn=max 0≤j≤nSj−Sn be a random walk Sn reflected in its maximum. Except in the trivial case when P(X≥0)=1, Rn 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 Rn 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 Sn is necessary for passage of Rn 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 Rn above linear and square root boundaries.
"Curve crossing for random walks reflected at their maximum." Ann. Probab. 35 (4) 1351 - 1373, July 2007. https://doi.org/10.1214/009117906000000953