Heavy-tailed Risk Variables And (or Weighted), Tail Asymptotic Behavior,

Since the 1960s, heavy-tailed distributions have been widely used in branching processes, queueing theory, risk theory including insurance and finance and other fields. Especially, heavy-tailed distributions are very important to insurance. For estate insurance, heavy-tailed distributions, such as subexponential distribution, have been regarded as standard model for individual counterclaim. Consequently, it is essential to study its characteristics and is help to insurance's progress. In the early researches on insurance and finance, the objects were supposed to be independent, identically distributed random variables. However, in the practical applications, there may exist some dependence among these random variables. And they may not be independent.In this paper, we still regard the heavy-tailed distributions as the main object and discuss tail asymptotics for the weighted sum of two heavy-tailed random variables. It consists of four chapters as follows:Chapter 1 introduces the notions and subclasses of heavy-tailed distribution, summarizes some results regarding the sum of heavy-tailed random variables, and elucidates what we will research in the paper.Chapter 2 discusses the weighted sum of random variables by copula function. In the paper we introduce the notions and functions of copula function, obtain some asymptotics for the weighted sum of two identically distributed random variables.Chapter 3 mostly discusses tail asymptotics for the weighted sum of regularly varying distributions. We obtain a property of tail probabilities of the weighted sum of two regularly varying random variables, simulate its result and another result refer to [36], using Monte Carlo's ways.In the last chapter, we sum up our conclusions, give a prospect of our study in the future.