Some Discussion Under Heavy-tailed Distribution

This dissertation consists of three chapters, which is completed during my MasterDegree of Science.Chapter One is a preface. It gives an introduction to the development and significanceof risk theory as well as probability. It also introduces the background and the mainideas of this dissertation.That is, we improve the condition that claims are independent inclassical risk models, discuss the asymptotic ruin probability of a continuous risk modeland more generally, explore the closure of product of dependent random variables inheavy-tailed distributions. In addition, some knowledge about heavy tails, copulae andtail-independence is presented at the end.Chapter Two investigates the ruin probability of a continuous time risk model underconstant interest rate with heavy tails. Based on the model in Chen and Ng(2007), wepropose an improved condition in which we allow general dependence structure amongindividual risks as long as they are upper tail independent.Under the assumption that theclaims is in the Extended regularly varying class, some simple asymptotic relations forthe finite and infinite ruin probability of our model are derived.In Chapter Three, we discuss the closure of product of dependent random variables inL. Under the assumption that random variables obey some kind of dependence structurein terms of Copulas, following Cline and Samorodnitsky(1994), we derive the condition ofY under which the product of XY is closed when X is in L and C(u₁, ..., u_n) is FGMCopulas.