  We study a new technique for the asymptotic analysis of heavy-tailed systems
conditioned on large deviations events. We illustrate our approach in the
context of ruin events of multidimensional regularly varying random walks. Our
approach is to study the Markov process described by the random walk
conditioned on hitting a rare target set. We construct a Markov chain whose
transition kernel can be evaluated directly from the increment distribution of
the associated random walk. This process is shown to approximate the
conditional process of interest in total variation. Then, by analyzing the
approximating process, we are able to obtain asymptotic conditional joint
distributions and a conditional functional central limit theorem of several
objects such as the time until ruin, the whole random walk prior to ruin, and
the overshoot on the target set. These types of joint conditional limit
theorems have been obtained previously in the literature only in the one
dimensional case. In addition to using different techniques, our results
include features that are qualitatively different from the one dimensional
case. For instance, the asymptotic conditional law of the time to ruin is no
longer purely Pareto as in the multidimensional case.
