  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 $(1, \, 3/2)$.
