  The contribution of this paper is to introduce change of measure based
techniques for the rare-event analysis of heavy-tailed stochastic processes.
Our changes-of-measure are parameterized by a family of distributions admitting
a mixture form. We exploit our methodology to achieve two types of results.
First, we construct Monte Carlo estimators that are strongly efficient (i.e.
have bounded relative mean squared error as the event of interest becomes
rare). These estimators are used to estimate both rare-event probabilities of
interest and associated conditional expectations. We emphasize that our
techniques allow us to control the expected termination time of the Monte Carlo
algorithm even if the conditional expected stopping time (under the original
distribution) given the event of interest is infinity -- a situation that
sometimes occurs in heavy-tailed settings. Second, the mixture family serves as
a good approximation (in total variation) of the conditional distribution of
the whole process given the rare event of interest. The convenient form of the
mixture family allows us to obtain, as a corollary, functional conditional
central limit theorems that extend classical results in the literature. We
illustrate our methodology in the context of the ruin probability $P(\sup_n S_n
>b)$, where $S_n$ is a random walk with heavy-tailed increments that have
negative drift. Our techniques are based on the use of Lyapunov inequalities
for variance control and termination time. The conditional limit theorems
combine the application of Lyapunov bounds with coupling arguments.
