Uniform asymptotics for a multidimensional renewal risk model with multivariate subexponential claims

In this paper, we investigate a multidimensional risk model driven by a common renewal process under a constant force of interest. The claim sizes generated by each line of business are independent and identically distributed random vectors with possibly dependent components, and their common distribution belongs to the class of multivariate subexponential distributions. We derive locally uniform asymptotic estimates for the probability that discounted aggregate claims enter certain ‘rare sets’, and extend these to uniform estimates over all time horizons under some extra mild conditions. As a direct application, we obtain uniform estimates for the finite-time ruin probability defined using various ruin sets. Additionally, we provide examples of distributions belonging to these multivariate heavy-tailed classes, which are not limited to the case of multivariate regular variation.