Background risk model in presence of heavy tails under dependence

Abstract: In this paper, we examine two problems on applied probability, which are directly connected with the dependence in presence of heavy tails. The first problem, is related to max-sum equivalence of the randomly weighted sums in bi-variate set up. Introducing a new dependence, called Generalized Tail Asymptotic Independence, we establish the bi-variate max-sum equivalence, under a rather general dependence structure, when the primary random variables follow distributions from the intersection of the dominatedly varying and the long tailed distributions. On base of this max-sum equivalence, we provide a result about the asymptotic behavior of two kinds of ruin probabilities, over a finite time horizon, in a bi-variate renewal risk model, with constant interest rate. The second problem, is related to the asymptotic behavior of the Tail Distortion Risk Measure, in a static portfolio, called Background Risk Model. In opposite to other approaches on this topic, we use a general enough assumption, that is based on multivariate regular variation.