Total Variation Approximation for Heavy-tailed Multidimensional Random Walks Given Ruin

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 characterized directly from the increment distribution of the associated random walk, and which is then 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. 1 Introduction The focus of this paper is the development of a precise asymptotic description of the distribution of a multidimensional regularly varying random walk (the precise meaning of which is given in Section 2) conditioned on hitting a rare target set represented as the union of half spaces. In particular, we develop tractable total variation approximations (in the sample path space), based on change-ofmeasure techniques, for such conditional stochastic processes. Using these approximations we are able to obtain, as a corollary, joint conditional limit theorems of speci…c objects such as the time until ruin, a Brownian approximation up to (just before) the time of ruin, the “overshoot”and the “undershoot”. The multidimensional problem is a natural extension of the classical one dimensional random ruin problem in a so-called renewal risk model for the insurance and queueing settings motivating this problem (c.f. the texts of [1, 2]). In this paper, we consider a d-dimensional regularly varying random walk S = (Sn : n 1) with S0 = 0 and drift 2 Rd so that ESn = n 6= 0 and de…ne TbA = inffn 0 : Sn 2 bA g; where A is the union of half spaces and points to the interior of some open cone that does not intersect A. [9] notes that P (TbA < 1) corresponds to the ruin probabilities for insurance companies with several lines of business. Using natural budget constraints related to the amount of money that can be transferred from one business line to another, it turns out that the target set takes precisely the form of the union of half spaces as we consider here. Our goal is to illustrate new techniques that can be used to describe very precisely the conditional distribution of the heavy-tailed processes given TbA < 1. Our approximations allow to obtain, in a relatively easy way, extensions of previous results in the literature obtained for the one dimensional setting. Asmussen and Klüppelberg ([3]) provide conditional limit theorems for the overshoot, the undershoot, and the time until ruin given the eventual occurrence of ruin. Similar