The Bivariate regular variation of randomly weighted sums revisited in the presence of interdependence

Abstract

We study the joint distribution of two randomly weighted sums. Inspired by the practical applications, we assume that the main random variables follow the non-standard bivariate regular variation, symbolically BRV , to put emphasis to the value of inhomogeneity of the risk distribution tails, while the random weights are weakly dependent with main random variables. Under some moment conditions on the random weights we show that the randomly weighted sums have BRV distribution with an analytic relation for the Radon measure, that captures the interdependence between the random weights and the main random variables. Under some stronger moments conditions, our result is extended uniformly, with respect to summands, covering also the case of infinite randomly weighted sums. In order to keep weak dependence structure among the random weights and the main random variables, we require the random weights to be independent each other, something that does not happen in models with insurance and financial risks. Up to recent years, such kind of approximations, even in one-dimensional case, had mostly theoretical interest, since underline the presence of (multivariate linear) single big jump principle. However, here we provide an application of the main results on ruin probability in a new flexible credit risk model. In our model, although we restrict ourselves to standard BRV , the obliged do not enter - quit necessarily simultaneously to the system, while the breach probability is not necessarily independent of the the amount of breach for the obliged. Finally, in the nonstandard BRV , with asymptotically dependent risks, we provide an application of the main results, to find the asymptotic behavior of a risk measure, which is called joint expected shortfall, that plays crucial role to the measure of the contagion of extreme risks