Some Properties Of Two Classes Of Dependence Structures With Applications In Heavy-tailed Risk Models

The risk theory provides a classical mathematical basis for the study of most insurance mathematical problems,while the core of the risk theory is the asymptotic approximation of the ruin probability of the risk model.At the same time,the dependence structure between risk variables is of great signif-icance to the characterization of risk models.Due to the frequent occurrence of extreme events in recent years,the heavy-tailed risk model has attracted a lot of attention.In the early,many scholars considered the risk variables are independent and identically distributed,such as Tang and Tsitsiashvili[35],Goovaerts et al.[16],and so on.However,with the development of cognition,the situation of independent distribution is obviously not consistent with the actual situation,therefore,the dependence structure between random vari-ables has attracted the attention of a large number of scholars.With the deep development of the dependent structure,the asymptotic approximation of the ruin probability of the risk model is getting closer and closer to the actual sit-uation.Therefore,the research about the dependent structure has important practical significance.This paper studied some properties of the two dependent structure which are the pairwise quasi-asymptotically independent structure(pQAI)and the pairwise tail quasi-asymptotically independent structure(pTQAI),then,we applied these properties to estimate asymptotic approximation in heavy-tailed risk model.In the first part of the second chapter,we studied the properties of the pairwise quasi-asymptotically independent structure.Let ｛Xi,i=1,…,n}be n real-valued random variables satisfy the pairwise quasi-asymptotically independent structure,{Wi,i=1,…,n} be another n nonnegative random variables independent of {Xi,i=1…,n},we can prove that W1X1,…,WnXn still satisfy the pairwise quasi-asymptotically independent structure under some conditions.Compared to Li[25],our result is greatly expand the ap-plication scope because it is in a general setting whether the primary random variables X1,…,Xn are heavy-tailed or not.In the second half part of this chapter,we applied the properties of the pQAI in a dependent discrete-time risk model,in this risk model,insurance risk and financial risk satisfying a wide range of dependent structure,then,we obtained the asymptotic approx-imation of the ruin probability in the risk model and improved the result in Yang et al.[44].In the first part of chapter 3,we studied the properties of the pairwise tail quasi-asymptotically independent structure.Let X1,X2,… be a sequence of real-valued and pTQAI random variables,g1,92,…be continuous and strictly increasing functions,I1,I2,… be non-empty,finite,and mutually disjoint index sets of positive integer,we can prove that Z1,Z2,… still satisfy the pairwise tail quasi-asymptotically independent structure under some conditions.Compared to Li[25],the conditions set in this paper are broader.In the second half part of this chapter,we consider a by-claim risk model with interest force,assuming that the main claim {Xi,i ≥1} and the corresponding by-claim{Yi,i≥ 1} satisfy the pairwise tail quasi-asymptotically independent structure,we obtained the asymptotic approximation of the ruin probability in the risk model.