  We propose a class of strongly efficient rare event simulation estimators for
random walks and compound Poisson processes with a regularly varying
increment/jump-size distribution in a general large deviations regime. Our
estimator is based on an importance sampling strategy that hinges on the
heavy-tailed sample path large deviations result recently established in Rhee,
Blanchet, and Zwart (2016). The new estimators are straightforward to implement
and can be used to systematically evaluate the probability of a wide range of
rare events with bounded relative error. They are "universal" in the sense that
a single importance sampling scheme applies to a very general class of rare
events that arise in heavy-tailed systems. In particular, our estimators can
deal with rare events that are caused by multiple big jumps (therefore, beyond
the usual principle of a single big jump) as well as multidimensional processes
such as the buffer content process of a queueing network. We illustrate the
versatility of our approach with several applications that arise in the context
of mathematical finance, actuarial science, and queueing theory.
