| It is well known that heavy-tailed distributions are widely used in research fields such as branching processes,queues,reliability theory,financial engineering,quantitative economics and insurance actuarial,and the Max-Sum equivalence of a sequence of heavy-tailed random variables in the independent case is a fundamental topic in applied probability theory.However,in real life,the assumption of independence is often untenable.In this paper,we will study the local Max-Sum equivalence of a sequence of heavy-tailed random variables under two types of special dependent structures.The research results of this paper can be applied to the local asymptotic characterization of ruin probability in risk theory and other fields.The full text is divided into five parts in total:In the first chapter,the background and significance of the Max-Sum equivalence-related problems and the current status of domestic and international research are first introduced,and give the definition of the Max-Sum equivalence.Then the research content and innovation points of this paper are given.In the second chapter,the definition and properties of the family of heavy-tailed distributions and its subfamilies are given first.Then the definitions and properties of the family of local long-tailed distributions and the family of local subexponential distributions are presented.In the third chapter,a brief introduction to the Farlie-Gumbel-Morgenstern(FGM)copula is given first.Then,when the real-valued random variable sequence satisfies the FGM copula,assuming that the random variables supports the local subexponential distribution,sufficient conditions for its local Max-Sum equivalence to hold are given in this case,and a theoretical proof of this conclusion is given.In the fourth chapter,we first give a concise introduction to Bernstein copula.Then,under the assumption that the random variables supports the local subexponential distribution,when the sequence of nonnegative heavy-tailed random variables supporting Bernstein copula,the local Max-Sum equivalence is given,and proved rigorously.Finally,on the basis of theoretical proof,stochastic simulations are further used to verify the validity of the theoretical results.In the fifth chapter,the work done in this paper is summarized,and further research topics to be studied are put forward. |
