Joint tail of randomly weighted sums under generalized quasi asymptotic independence

TL;DR

This paper investigates the joint tail behavior of randomly weighted sums in two dimensions, considering dependence structures and heavy-tailed distributions, with applications to risk models and ruin probabilities.

Contribution

It introduces a new dependence structure combined with heavy-tailed classes and a novel approach to two-dimensional regular variation aligned with the big jump principle.

Findings

New dependence structure for 2D heavy-tailed distributions

Closure properties of the introduced classes

Application to finite-time ruin probability in risk models

Abstract

In this paper we revisited the classical problem of max-sum equivalence of randomly weighted sums in two dimensions. In opposite to the most papers in literature, we consider that there exists some interdependence between the primary random variables, which is achieved by a combination of a new dependence structure with some two-dimensional heavy-tailed classes of distributions. Further, we introduce a new approach in two-dimensional regular varying distributions, that in contrast to well-established multivariate regularly varying distributions, is consistent with the multivariate non-linear single big jump principle. We study some closure properties of this, and of other two-dimensional classes. Our results contain the finite-time ruin probability in a two-dimensional discrete time risk model