The Asymptotic Behavior Of Tail Probability For Randomly Weighted Sums With Heavy-Tailed Increments Under Dependence Structure And Its Applications

In recent years, the extreme events occurred occasionally in nature. For example, the 2004 Indian Ocean tsunami, the 2005 Katrina Hurricane, the 2008 Wenchuan earthquake, the 2010 Haiti earthquake, the 2010 Chile earthquake, the 2015 flood of America and so on. These extreme events often lead to heavy-tailed claim amounts. Though the occurring probability of each extreme event is small, it would bring a huge impact to the insurance company as long as it happened, and even result in bankruptcy of the company. Historical data shows that there are obvious deviations to characterize the extreme claims by classically light-tailed distributions. The scholars of applied probability show that the heavy-tailed distributions are suitable for these requirements. Therefore, the study of risk model with heavy-tailed claims has attracted more and more attention, and become a hot topic in actuarial science.The model of random weighted sums is the most basic one in actuarial sci-ence. The corresponding results of the tail probability can be used to estimate the ruin probability of the insurer. Therefore, the study of the tail proba-bilities for randomly weighted sums and their maxima has attracted much attention in the literature of applied probability and actuarial science. Under heavy-tailed claim cases, this problem mainly focuses on characterizing the asymptotic behavior of tail probabilities for random weighted sums and their maxima distributions. This paper will investigate the asymptotic behavior of tail probabilities for randomly weighted sums and their maxima with its appli-cation in risk management, where the risk variables are dependent according to a wide type of dependence structure.The main content of this present paper includes the following two aspects. The first is that we give explicit lower and upper bounds for the tail probabil-ities of random weighted sums and their maxima under some mild conditions on the right tails of the weights, in the case where the risk variables are depen-dent according to a wide type of dependence structure and belong to the class of dominated variation, but are independent of the random weights. Based on the former results, we establish asymptotic equivalence formulas for the tail probabilities of two risk variables which belong to the long-tailed and subex-ponential class, respectively. Therefore, our results in this paper improve the ones of Yang et al. (2014) and Cheng (2014) substantially.The second is that we consider a financial risk model, in which the loss variables belong to extended-regularly varying class, and the discount factor variables are dependent according to a wide type of dependence structure. We aim at discussing the risk measure CTE (Condition Tail Expectation). Under some mild conditions, we establish some asymptotic (weak) equivalence formulas for the CTE. Our obtained result in this paper extends the one of Yang et al. (2015).