Total variation approximations and conditional limit theorems for multivariate regularly varying random walks conditioned on ruin

Abstract

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.