Uniform asymptotics for a multidimensional renewal risk model with random number of delayed claims and multivariate subexponentiality

Abstract: In this paper we examine a multivariate risk model, with common renewal counting process, constant interest rate, and each claim vector is accompanied by a random number of delayed claim vectors. The interest is focused on the asymptotic behavior of the entrance probability of the discounted aggregate claims into some rare-sets, over a finite and an infinite time horizon. Our results study the the case where the main claims and the delayed claims have in some sense, asymptotic equivalent tails, but also the case where the delayed claims are negligible with comparisons with the main claims. More precisely, our estimations over finite time horizon are equipped with local uniformity, and are valid under the assumption of multivariate subexponential distributions for the claim distributions. On the case of infinite time horizon we need a mild restriction on the distribution class of multivariate subexponential distributions with positive lower Karamata index. The asymptotic relations reflect completely as all the sources of randomness, under the concrete rare-sets A, and the different dependence structures as well, without loosing elegance in spite of their generality. Further, we provide some more explicit formulas, together with relaxations of some assumptions, for the claim distributions from the multivariate regular variation. For the proof of the main results on infinite time case and for the construction of examples of multivariate distributions we need some closure properties of subexponential distributions with positive lower Karamata index. Especially, we present some necessary and sufficient conditions for the closure property with respect to convolution and some sufficient conditions for the closure property with respect to product convolution. Finally, we carry out some numerical studies to show the accuracy of our asymptotic estimations.

• The paper establishes uniform asymptotic approximations for discounted aggregate claims in a multidimensional setting by modeling both immediate and delayed claims.
• It demonstrates a dichotomy where comparable tail behavior of delayed and main claims leads to significant contributions, while negligible tails render delayed claims asymptotically invisible.
• The study extends heavy-tailed risk analysis beyond regular variation, using multivariate subexponential classes and numerical verification to support its theoretical advances.

Uniform Asymptotics for Multidimensional Renewal Risk with Delayed Claims and Multivariate Subexponentiality

Model Framework and Problem Statement

The paper investigates the entrance probability asymptotics for discounted aggregate claims in a multidimensional renewal risk setting involving delayed claims and multivariate subexponential distributions. The model treats an insurer with $d$ lines of business. Claims are induced by a renewal process capturing arrival times $\tau_i$, with each main claim vector $\mathbf{X}^{(i)} \in \mathbb{R}^d_+$ at time $\tau_i$ possibly generating $M_i$ additional delayed claim vectors $\mathbf{Y}^{(i, j)}$ at random delayed times $\tau_i + D_{ij}$. The discounted aggregate claims process over horizon $t$, given deterministic interest rate $r\geq 0$, is:

$${\bf D}_r(t) = \sum_{i=1}^{N(t)} {\bf X}^{(i)} e^{-r \tau_i} + \sum_{i=1}^{N(t)} \sum_{j=1}^{M_i} {\bf Y}^{(i, j)} e^{-r (\tau_i + D_{ij})} \mathbf{1}_{\{\tau_i + D_{ij} \leq t\}}$$

The core object of study is $\tau_i$0 and $\tau_i$1 as $\tau_i$2, for rare set $\tau_i$3 from the family $\tau_i$4 (open, increasing, convex-complement sets in $\tau_i$5 not containing $\tau_i$6), pertinent for high-loss/ruin scenarios. The analysis is performed both on finite and infinite time horizons, with emphasis on the uniformity and generality of the asymptotics.

Multivariate Heavy-Tailed and Subexponential Classes

The research develops its asymptotic characterizations within the framework of multivariate subexponentiality $\tau_i$7 and its restriction $\tau_i$8, defined via large deviation scaling of the distributions $\tau_i$9 (main claims) and $\mathbf{X}^{(i)} \in \mathbb{R}^d_+$0 (delayed claims). Crucially, these classes encompass MRV laws but also heavier and mod-erate heavy-tailed marginals (e.g., lognormal), thereby extending applicability beyond models based solely on regular variation.

For any $\mathbf{X}^{(i)} \in \mathbb{R}^d_+$1, the threshold variable $\mathbf{X}^{(i)} \in \mathbb{R}^d_+$2 is central. $\mathbf{X}^{(i)} \in \mathbb{R}^d_+$3 iff $\mathbf{X}^{(i)} \in \mathbb{R}^d_+$4 is classic univariate subexponential; $\mathbf{X}^{(i)} \in \mathbb{R}^d_+$5 iff $\mathbf{X}^{(i)} \in \mathbb{R}^d_+$6 is subexponential with strictly positive lower Karamata index, a condition required for many infinite sum asymptotics.

Main Theoretical Results

1. Local Uniform Asymptotics on Finite Horizon

For $\mathbf{X}^{(i)} \in \mathbb{R}^d_+$7, under independence Assumptions and light-tailed multiplicities for $\mathbf{X}^{(i)} \in \mathbb{R}^d_+$8, the following dichotomy governs local uniform asymptotic behavior (Theorem 3.1):

• Case (i): Equivalent tails ($\mathbf{X}^{(i)} \in \mathbb{R}^d_+$9), $\tau_i$0:

$\tau_i$1

uniformly for $\tau_i$2.

Both main and delayed claims contribute at first order, establishing the relevance of delayed claims whenever their marginal tails are commensurate with the main claims.

• Case (ii): Negligible delayed claim tails ($\tau_i$3):

$\tau_i$4

The aggregate risk behaves as if delayed claims did not exist—a formal statement that under heavy-tailed asymptotics, delayed but light-tailed claim vectors are asymptotically invisible in rare event regimes.

Explicit forms for sets $\tau_i$5 of practical relevance (such as $\tau_i$6, $\tau_i$7) are established using regression dependence and marginal closure properties, providing reduced asymptotic forms as sums of marginal tail probabilities.

2. Infinite Horizon Asymptotics

For $\tau_i$8 (strictly positive rate required for non-defective limiting distribution), and with delayed claims potentially having moderate or light tails, Theorem 4.1 states:

• Case (i): $\tau_i$9, $M_i$0:

$M_i$1

• Case (ii): $M_i$2, $M_i$3, $M_i$4:

$M_i$5

Again, negligible delayed-claim tails lead to effective asymptotics uniquely determined by the main claim process.

For the $M_i$6 subclass $M_i$7, $M_i$8, explicit expressions involving the renewal Laplace functional and tail measures are supplied:

$M_i$9

Closure Properties and Technical Contributions

A salient technical accomplishment is the characterization of closure properties for $\mathbf{Y}^{(i, j)}$0 and $\mathbf{Y}^{(i, j)}$1 under convolution and random scaling (“product convolution”). Necessary and sufficient conditions—especially for product convolution—enable the construction of new multidimensional subexponential distributions and facilitate proofs of the main results. Notably, the work demonstrates that the closure (and thus the asymptotic structure) can be reduced to appropriate univariate analogs built on $\mathbf{Y}^{(i, j)}$2.

The paper also presents a series of examples illustrating distributions in $\mathbf{Y}^{(i, j)}$3 and $\mathbf{Y}^{(i, j)}$4 outside the MRV regime, resulting in broader generality.

Numerical Verification and Moderate Heavy Tails

Numerical experiments corroborate the sharpness of the asymptotic formulas, including in scenarios where marginals are moderately heavy-tailed (e.g., lognormal), a regime often avoided in previous actuarial literature due to slow convergence. The local and global uniformity assertions are sustained by these studies.

Theoretical and Practical Implications

This research establishes precise conditions under which delayed claims impact (or do not impact) rare event probabilities in risk models with multiple portfolios and complex dependence, when claim vector distributions are subexponential in a multivariate sense. The results extend the range of tractable models to those with moderate and non-MRV heavy tails, unify approaches to finite- and infinite-time rare event probabilities under renewal arrivals, and resolve longstanding questions regarding the interplay of tail-heaviness and dependence in delayed claim risk models.

Practically, these findings inform capital requirement and solvency calculations in lines of business where tail risk is not always asymptotically Pareto, and claim delays are both stochastic in number and size. The identification of “asymptotic invisibility” regimes for delayed claims is highly relevant for actuaries conducting risk aggregation across heterogeneous portfolios.

Theoretically, the analysis reinforces the principle that in multivariate heavy-tailed settings, rare-event probabilities are insensitive to numerous modeling details, except the extremal behavior and relative tail indices of the underlying claim/distribution components (a "multivariate linear single big jump" principle). Future work may explore non-renewal arrival processes, random investment returns, or further relaxations in dependence structures, supported by the closure properties established in this paper.

Conclusion

The paper systematically determines uniform, explicit asymptotic approximations for rare-event probabilities in a multidimensional renewal risk context with delayed claims, within the framework of multivariate subexponentiality. The research makes critical advances both in the breadth of applicable multivariate heavy-tailed models and in the analytic tools facilitating such advances, with clear implications for insurance risk theory, extreme value analysis, and applied probability.