  Let $R_n=\max_{0\leq j\leq n}S_j-S_n$ be a random walk $S_n$ reflected in its
maximum. Except in the trivial case when $P(X\ge0)=1$, $R_n$ 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.
