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

TL;DR

The paper develops a unified framework for multivariate renewal risk models with common claim vectors drawn from heavy-tailed, multivariate subexponential classes. By introducing sets $A$ in the positive orthant and the discounted aggregate claims ${\bf D}_r(t)$, it derives local and global uniform asymptotics for the entrance probability into $xA$, expressed as $P[ {\bf D}_r(t)\in xA] \sim \int_0^t P[ {\bf X}\in x e^{rs}A]\lambda(ds)$, and shows corresponding implications for finite-time and infinite-time ruin probabilities. The results hold under the multivariate subexponential classes $\mathcal{S}_A$ and $\mathcal{A}_A$, with extensions to MRV and non-standard MRV settings, and include concrete distribution constructions illustrating how various dependence structures yield the required tail behavior. The findings enable robust assessment of ruin risks across multiple lines of business under heavy-tailed joint claims, beyond the restrictive multivariate regular variation framework. Overall, the work advances uniform asymptotic techniques in multidimensional risk theory and provides practical tools for finite-horizon ruin analysis with multivariate heavy tails.

Abstract

In this paper, we study a multidimensional risk model with a common renewal process and in the presence of a constant interest force. The claim sizes are independent and identically distributed random vectors, with the distribution of dependent components belonging to the class of multivariate subexponential distributions. We establish locally uniform asymptotic estimations for the entrance probability of the discounted aggregate claims into some rare sets, and further derive asymptotic estimations uniformly over all the time horizons. Furthermore, we present some distribution examples that belong to these multivariate heavy-tailed distribution classes, which are not restricted only to the case of multivariate regular variation.

Theorems & Definitions (18)

• Theorem 3.1
• Theorem 3.2
• Remark 3.1
• Remark 3.2
• Remark 3.3
• Remark 3.4
• Corollary 3.1
• Remark 3.5
• Example 4.1
• Remark 4.1