Joint tail of randomly weighted sums under generalized quasi asymptotic independence
Dimitrios G. Konstantinides, Charalampos D. Passalidis
TL;DR
The work addresses joint tail behavior of randomly weighted sums in a two-dimensional setting under generalized dependence (GTAI and GQAI). It introduces a two-dimensional regular variation framework $\mathcal{R}_{(-\alpha_1,-\alpha_2)}^{(2)}$ and related dependence classes to capture non-linear single-big-jump phenomena, then derives asymptotic equivalences for $\mathbf{P}[S_n^{\Theta}>x, T_m^{\Delta}>y]$ and related ruin probabilities. It establishes closure properties of two-dimensional tail classes under sums and random-weight convolution, enabling robust transfer of tail results across operations. The results apply to finite-time ruin probabilities in a two-dimensional risk model with stochastic discount factors and provide moment-assisted weighted-tail formulas when heavier-tailed weight distributions are present.
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