Asymptotic Tail Probabilities Research Articles

Asymptotic Tail Probability of the Discounted Aggregate Claims under Homogeneous, Non-Homogeneous and Mixed Poisson Risk Model

In this paper, we derive a closed-form expression of the tail probability of the aggregate discounted claims under homogeneous, non-homogeneous and mixed Poisson risk models with constant force of interest by using a general dependence structure between the inter-occurrence time and the claim sizes. This dependence structure is relevant since it is well known that under catastrophic or extreme events the inter-occurrence time and the claim severities are dependent.

Asymptotic Behavior of Discounted Aggregate Claims in a Nonstandard Two-dimensional Risk Model with Dependent Claim Sizes

In this article, we study a nonstandard bivariate renewal risk model with dependent claim sizes, where two kinds of claim sizes constitute a sequence of independent and identically distributed random vectors satisfying a certain dependence structure, two claim-number processes are not necessarily independent. When the marginal distributions of the random vectors belong to the intersection of the class of long-tailed distributions and the class of the dominated variation distributions, we obtain joint tail asymptotic formulas of discounted total claims. The result shows that the dependence between two kinds of claims has an effect on the joint asymptotic tail probability of discounted total claims.

Asymptotic tail probability of weighted infinite sum of conditionally dependent and consistently varying tailed random variables

This paper investigates the asymptotic behavior of the tail probability of a weighted infinite sum of random variables with consistently varying tails under two conditional dependence structures. The obtained results extend and improve the existing results of Bae and Ko (J. Korean Stat. Soc. 46:321–327, 2017).

Normal Inverse Gaussian Approximation for Arrival Time Difference in Flow-Induced Molecular Communications

In this paper, we consider molecular communications in one-dimensional flow-induced diffusive channels with a perfectly absorbing receiver. In such channels, the random propagation delay, until the molecules are absorbed, follows an inverse Gaussian (IG) distribution and is referred to as first hitting time. Knowing the distribution for the difference of the first hitting times of two molecules is very important if the information is encoded by a limited set of molecules and the receiver exploits their arrival time and/or order. Hence, we propose a moment matching approximation by a normal IG (NIG) distribution and we derive an expression for the asymptotic tail probability. Numerical evaluations show that the NIG approximation matches very well with the exact solution obtained by numerical convolution of the IG density functions. Moreover, the asymptotic tail probability outperforms state-of-the-art tail approximations.

Aggregation of randomly weighted large risks

Asymptotic tail probabilities for linear combinations of randomly weighted order statistics are approximated under various assumptions. One key assumption is the asymptotic independence for all risks. Therefore, it is not surprising that the maxima represents the most influential factor when one investigates the tail behaviour of our considered risk aggregation, which, for example, can be found in the reinsurance market. This extreme behaviour confirms the ‘one big jump’ property that has been vastly discussed in the existing literature in various forms whenever asymptotic independence is present. An illustration of our results together with a specific application are explored under the assumption that the underlying risks follow the multivariate log-normal distribution.

Randomly Weighted Sums of Pairwise Quasi Upper-Tail Independent Increments with Application to Risk Theory

This article investigates the asymptotic tail probabilities of randomly weighted sums and their maximum of pairwise quasi upper-tail independent increments with non negative and arbitrarily dependent random weights. An application of the obtained results to risk theory is proposed.

Exact Tail Asymptotics of the Total Loss of Largest Claim Reinsurance Treaties

Asymptotic tail probabilities for linear combinations of randomly weighted order statistics are approximated under various assumptions. One key assumption is the asymptotic independence for all risks. Therefore, it is not surprising that the maxima represents the most influential factor when investigates the tail behaviour of our considered risk aggregation, which for example, can be found in the reinsurance market. This extreme behaviour confirms the one big jump property that has been vastly discussed in the existing literature in various forms whenever the asymptotic independence is present. An illustration of our results together with a specific application are explored under the assumption that the underlying risks follow the multivariate Log-normal distribution.

Uniform asymptotics for the finite-time and infinite-time ruin probabilities in a dependent risk model with constant interest rate and heavy-tailed claims

We consider a nonstandard risk model with constant interest rate. For the case where the claim sizes follow a common heavy-tailed distribution and fulfill a dependence structure proposed by Geluk and Tang [J. Geluk and Q. Tang, Asymptotic tail probabilities of sums of dependent subexponential random variables, J. Theor. Probab., 22:871–882, 2009] while the interarrival times fulfill the so-called widely lower orthant dependence, we establish a weakly asymptotically equivalent formula for the infinite-time ruin probability. In particular, when the dependence structure for claim sizes is strengthened to the widely upper orthant dependence, this result implies a uniformly asymptotically equivalent formula for the finite-time and infinite-time ruin probabilities.

Subexponential tails of discounted aggregate claims in a time-dependent renewal risk model

Consider a continuous-time renewal risk model with a constant force of interest. We assume that claim sizes and interarrival times correspondingly form a sequence of independent and identically distributed random pairs and that each pair obeys a dependence structure described via the conditional tail probability of a claim size given the interarrival time before the claim. We focus on determining the impact of this dependence structure on the asymptotic tail probability of discounted aggregate claims. Assuming that the claim size distribution is subexponential, we derive an exact locally uniform asymptotic formula, which quantitatively captures the impact of the dependence structure. When the claim size distribution is extended regularly varying tailed, we show that this asymptotic formula is globally uniform.

Subexponential tails of discounted aggregate claims in a time-dependent renewal risk model

Consider a continuous-time renewal risk model with a constant force of interest. We assume that claim sizes and interarrival times correspondingly form a sequence of independent and identically distributed random pairs and that each pair obeys a dependence structure described via the conditional tail probability of a claim size given the interarrival time before the claim. We focus on determining the impact of this dependence structure on the asymptotic tail probability of discounted aggregate claims. Assuming that the claim size distribution is subexponential, we derive an exact locally uniform asymptotic formula, which quantitatively captures the impact of the dependence structure. When the claim size distribution is extended regularly varying tailed, we show that this asymptotic formula is globally uniform.

Extremes on the discounted aggregate claims in a time dependent risk model

This paper presents an extension of the classical compound Poisson risk model for which the inter-claim time and the forthcoming claim amount are no longer independent random variables (rv's). Asymptotic tail probabilities for the discounted aggregate claims are presented when the force of interest is constant and the claim amounts are heavy tail distributed rv's. Furthermore, we derive asymptotic finite time ruin probabilities, as well as asymptotic approximations for some common risk measures associated with the discounted aggregate claims. A simulation study is performed in order to validate the results obtained in the free interest risk model.

Erratum to: Asymptotic tail probability of randomly weighted sums of dependent random variables with dominated variation

This paper investigates the asymptotic behavior of tail probability of randomly weighted sums of dependent and real-valued random variables with dominated variation, where the weights form another sequence of nonnegative random variables. The result we obtain extends the corresponding result of Wang and Tang[7].

Asymptotic tail probability of randomly weighted sums of dependent random variables with dominated variation

Asymptotic tail probability of randomly weighted sums of dependent random variables with dominated variation

Maxima of moving sums in a Poisson random field

In this paper we examine the extremal tail probabilities of moving sums in a marked Poisson random field. These sums are computed by adding up the weighted occurrences of events lying within a scanning set of fixed shape and size. We also provide an alternative representation of the constants of the asymptotic formulae in terms of the occupation measure of the conditional local random field at zero, and extend these representations to the constants of asymptotic tail probabilities of Gaussian random fields.

Maxima of moving sums in a Poisson random field

In this paper we examine the extremal tail probabilities of moving sums in a marked Poisson random field. These sums are computed by adding up the weighted occurrences of events lying within a scanning set of fixed shape and size. We also provide an alternative representation of the constants of the asymptotic formulae in terms of the occupation measure of the conditional local random field at zero, and extend these representations to the constants of asymptotic tail probabilities of Gaussian random fields.

Dependence and the asymptotic behavior of large claims reinsurance

We consider an extension of the classical compound Poisson risk model, where the waiting time between two consecutive claims and the forthcoming claim are no longer independent. Asymptotic tail probabilities of the reinsurance amount under ECOMOR and LCR treaties are obtained. Simulation results are provided in order to illustrate this.

Asymptotic Tail Probabilities for Large Claims Reinsurance of a Portfolio of Dependent Risks

We consider a dependent portfolio of insurance contracts. Asymptotic tail probabilities of the ECOMOR and LCR reinsurance amounts are obtained under certain assumptions about the dependence structure.

Asymptotic Tail Probabilities for Large Claims Reinsurance of a Portfolio of Dependent Risks

We consider a dependent portfolio of insurance contracts. Asymptotic tail probabilities of the ECOMOR and LCR reinsurance amounts are obtained under certain assumptions about the dependence structure.

Insensitivity to Negative Dependence of Asymptotic Tail Probabilities of Sums and Maxima of Sums

Let {X 1, X 2,…} be a sequence of pairwise negatively dependent random variables with distributions F 1, F 2,… on (− ∞, ∞), and let S (n) be the maximum of its first n partial sums S 1,…, S n . This article studies the asymptotic tail probabilities of S n and S (n). Under suitable regularity conditions on the distributions F 1,…, F n , we prove that both P(S n > x) and P(S (n) > x) are asymptotic to as x → ∞, indicating that the negative dependence among the random variables does not affect the asymptotic tail behavior of S n and S (n). For a special case where the sequence {X 1, X 2,…} is generated by pairwise negatively dependent and identically distributed random variables weighted by positive constants fulfilling a certain summability condition, we further prove that the asymptotic relations obtained are uniform for all n = 1,2,….

Sums of Dependent Nonnegative Random Variables with Subexponential Tails

In this paper we study the asymptotic tail probabilities of sums of subexponential, nonnegative random variables, which are dependent according to certain general structures with tail independence. The results show that the subexponentiality of the summands eliminates the impact of the dependence on the tail behavior of the sums.