Tail behavior of randomly weighted sums with interdependent summands
Dimitrios G. Konstantinides, Remigijus Leipus, Charalampos D. Passalidis, Jonas Siaulys
TL;DR
This work investigates the tail behavior of randomly weighted sums when there is interdependence between the primary variables and the random weights, extending max-sum tail equivalence to a broad dependent framework. It introduces a novel dependence structure via functions $g_{ij}$ that modulate joint tails conditionally on weights, and proves that $\mathbf{P}[S_n^{\Theta}>x]$ is asymptotically equivalent to sums of individual tails $\mathbf{P}[\Theta_i X_i>x]$ under $pTAI$ or $pQAI$, with sharper forms for regularly varying components. The results yield finite-time ruin probabilities in discrete-time risk models and extend to generalized moments, offering bounds and asymptotics for risk measures such as expected shortfall, even when weights and summands are dependent. The paper further extends to randomly weighted and stopped sums, including random horizons, establishing asymptotics of $\mathbf{P}[S_N^{\Theta}>x]$ and related ruin quantities under identically distributed components, with tail classes closed under these operations. Overall, the work broadens the applicability of tail-asymptotic methods to interdependent weighted sums, with direct implications for risk management and actuarial science.
Abstract
We reconsider a classical, well-studied problem from applied probability. This is the max-sum equivalence of randomly weighted sums, and the originality is because we manage to include interdependence among the primary random variables, as well as among primary random variables and random weights, as a generalization of previously published results. As a consequence we provide the finite-time ruin probability, in a discrete-time risk model. Furthermore, we established asymptotic bounds for the generalized moments of randomly weighted sums in the case of dominatedly varying primary random variables under the same dependence conditions. Finally, we give some results for randomly weighted and stopped sums under similar dependence conditions, with the restriction that the random weights are identically distributed, and the same holds for the primary random variables. Additionally, under these assumptions, we find asymptotic expressions for the random time ruin probability, in a discete-time risk model.