Precise Large Deviations For A Dependent Risk Modelunder Heavy-tailed Claims

Large deviations theory is an important tool to study the ruin probability, so the study of large deviations becomes a focus in the field of financial insurance. Extreme events may occur in the financial insurance industry, such as earthquakes, tsunamis and so on. Heavy-tailed distributions can portray large claims, so the study of large deviations under heavy-tailed will be of great significance. In addition to considering the phenomenon of heavy-tailed distribution, the dependence between risks is also needed to be considered. In practical problems, the situations that the individual risks which are not independent each other is often came across, such as widely dependence or other dependencies. Thus, the study of precise large deviations for a dependent risk model under heavy-tailed claims has practical value and significance.Taking the class D(40) L to the main object, the precise large deviations of dependent risk model is discussed. The main contents are as follows:Firstly, the precise large deviations of the random variables which is widely dependent but distributed differently is investigated. {, 1}kX k ≥ are not-negative widely dependence random variables with distribution functions {, 1}kF k ≥. Under certain conditions, the uniformly asymptotic results for the corresponding random sums are obtained, thus some conclusions of existing literature are further generalized.Secondly, on the basis of the previous section, customer-arrival-based risk model is proposed. The large deviations of loss process in the model is studied, which {, 1}kB k ≥ from negative dependence(NOD) generalized to widely dependence(WOD) and the class C extended to the class D(40) L.Finally, the model of delayed-claims risk is studied. Assuming that the main claim amount and delayed-claim amount are sequence of the random variables that asymptotically independent and identically distributed, precise large deviations of the loss process are received under the condition that the claims belongs to the class D(40) L. Then based on the relationship of several dependence structures, the corresponding conclusions of the precise large deviations are obtained.