Studies Of Some Probability Problems Associated With Heavy-tailed Risk

Since the risk of large claims has a great impact on the insurance company,the risk theory about extremal events has attracted more and more attention.In mathematics, this type of risk can be modeled by some random variable with a heavy-tailed distribution. The class of heavy-tailed distributions include a lot of important subclasses,one of which is the so-called subexponential distribution class,which covers a large range of distributions and could characterize the heavy-tailed risk very well.The subexponential distribution was first proposed by Chistyakov,V.P.[14]in 1964 and was applied to branching processes,and then it was applied rapidly to many fields such as risk theory, queuing theory,etc.At the same time,many literatures were devoted to the properties of subexponential distributions.This dissertation discusses some probability problems connected with heavy-tailed risk and concerns the following three topics:the properties of subexponential distributions ang related classes and their applications,the theory of large deviations and the generalized renewal measures.As preliminaries,the beginning part in this dissertation gives an explanation for some notations,classes of functions and classes of heavy-tailed distributions.In the first chapter,we discuss some properties of subexponential distributions and their related classes with their applications.Asmussen[4]introduced the class of local subexponential distributions.We first prove the non-closure under convolution of the local subexponential family.Next,we obtain some equivalent assertions about the local behavior of the tail probability of the maximum of random walks with negative drifts. Moreover,Remark 4.2 of Shimura and Watanabe[56]illustrated by a counterexample that Lemma 2.1(ⅳ) of Cline[17]is incorrect,which affects the validity of some related results. But according to Pakes[51],there exists a shortcoming in Shimura and Watanabe's counterexample, hence we first propose a further counterexample.What's more,we show that Corollary 3.2(ⅰ) of Cline[17]is wrong itself.Finally,we establish a result about the subexponentiality of the minimum of two random variables.In the second chapter,we first establish the theory of the second order subexponential distributions with finite means and then obtain a second order approximation to the tail of a compound distribution.The class of second order subexponential distributions defined in this paper can be viewed as a extension of the related class defined by Geluk[26]. Our results also generalize the correspondent results of Omey and Willekens[48]without assuming the existence of a density function of the underlying distribution.Next,we apply these results to risk theory and obtain the second order asymptotics for the ruin probabilities in renewal risk model,whereas in classical results,the related results are restricted to the Cramér-Lunderberg model. In the third chapter,we discuss the large deviation probability for the random sums of heavy-tailed non-negative i.i.d,random variables.The related result in this respect was first established by Klüppelberg and Mikosch[36].But their condition was too restrictive and then was reduced by Tang et al.[58].Furthermore,we establish the necessary and sufficient condition for this type of large deviation relations.What' more,we find that this condition holds not only for i.i.d,random variables but also for non-independent or non-identically distributed random variables.In the fourth chapter,we obtain a one-sided large deviation local limit theorem for the sums of heavy-tailed non-negative i.i.d,random variables.Our result improves the related results in Baltrūnas[5]because we establish such a result assuming only a onesided condition on the tail of the underlying distribution.Thus,the correspondent result of Doney[19]is generalized thoroughly.In the last chapter,we present all sorts of Blackwell-type renewal theorems by making the connection between Blackwell renewal theorem for renewal measures and that for generalized renewal measures.We show that different types of Blackwell-type renewal theorems will be established according to which of the tails of weighted renewal constants or the underlying distribution is asymptotically heavier.