A Dichotomy for Sampling Barrier-Crossing Events of Random Walks with Regularly Varying Tails

We study how to sample paths of a random walk up to the first time it crosses a fixed barrier, in the setting where the step sizes are iid with negative mean and have a regularly varying right tail. We introduce a desirable property for a change of measure to be suitable for exact simulation. We study whether the change of measure of Blanchet and Glynn (2008) satisfies this property and show that it does so if and only if the tail index $\alpha$ of the right tail lies in the interval $(1, \, 3/2)$.