Asymptotic Approximation Of Tail Probabilities For Heavy-tailed Risk Models Under Two Types Of Dependence Structure

In actuarial science,risk theory has always been a hot topic.And one of the most important issues in risk theory is the asymptotic estimation of ruin probability.In fact,the estimation of ruin probability of risk models is tail probability of the maximum of the present value variables of claim.On the other hand,in consideration of the fact that the ruin of many finance and insurance company in reality is caused by some extreme events,a common property of large claims caused by extreme events is heavy-tailed.Hence,recently,more and more applied probability researchers have paid their much attention in heavy-tailed distributions and heavy-tailed risk models.Based on this,this paper mainly investigates the asymptotic estimation of the present value variables of claims and their maxima when it is assumed that the claims are heavy-tailed and some dependent structures are allowed among the claims.The main content of this paper includes the following parts:The first two chapters mainly introduce the research background and explain the basic symbols,concepts and conventions involved in the paper.Firstly,the research background of this paper is introduced,and then the definition of the heavy-tailed distributions and some dependent structures are introduced.The third chapter considers a nonstandard renewal risk model,in which the claim sizes form a sequence of nonnegative LWQD*(linearly wide quadrant dependent*)random variables with common distribution belonging to S*(strong subexponential distribution),and their inter-arrival times constitute another sequence of nonnegative,independent and identically distributed random variables,but independent of the claim sizes.By establishing a Kestentype inequality for LWQD*randomly weighted sums,this paper achieves a uniformly asymptotic estimation for the tail probability of discounted aggregate claim process.The obtained result extends the ones in Chen[10].The fourth chapter considers the asymptotic estimation of tail probability for randomly weighted sums and their maxima.Let {Xn,n?1} be a sequence of real-valued LWQD(linearly wide quadrant dependent)random variables(always denotes claims)with common distribution belonging to S*(strong subexponential distribution),and {?n,n?1} be another sequence of nonnegative and arbitrarily dependent random variables(always denotes discounted factors),but independent of the claim sizes.This paper firstly obtains the asymptotic estimation of tail probability for randomly weighted sums and their maxima under the assumption that {?n,n?1} are bounded,which extends Theorem 4.1 of Yu et al.[50].Furthermore under some mildly technical conditions,this paper also derives the asymptotic estimation of tail probability for randomly weighted sums and their maxima when {?n,n?1} can take unbounded values.This result extends Theorem 2.3 of Yu and Cheng[49].Finally,an application of our main results to risk theory is also proposed in this paper.