Curve crossing for random walks reflected at their maximum

TL;DR

This paper investigates the conditions under which a reflected random walk crosses various curved boundaries, extending classical results to power law, linear, and square root boundaries, and analyzing the influence of the tail heaviness of increments.

Contribution

It provides necessary and sufficient conditions for the finiteness of passage times of the reflected random walk above curved boundaries, including power law, linear, and square root cases.

Findings

Passage over any height occurs with probability 1 if not trivial.

Conditions depend on the heaviness of the negative tail of increments.

Finiteness of expected passage time is characterized for linear and square root boundaries.

Abstract

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.