Multivariate Convolution Equivalent Distribution And Their Applications

Subexponential distribution class S is a common kind of heavy tail distribution class,from Chistyakov[26](1964)according to Feller[11](1941)to study the existence of the solution of the updated equation and Bellman Harris[1](1952),Levinson[17](1960)and Vinogradov[23](1964)are proposed on the basis of studying the form of the solution of the updated equation respectively.Chover[7](1973)studying the update function,the convolution equivalent distribution cluster S(?)(??0)(S-S(0))is proposed,which is the generalization of the subexponential distribution class S,and many scholars later study the properties of S(?)class,And the application of many aspects is given.The different concepts of multivariate subexponential distribution are presented by Cline and Resnick[6](1992),Omey[18](2006)and Samorodnitsky and Sun[20](2016)respectively,and their properties are studied.Many of these properties are natural popularization of the properties of S class.Among them,the concept of the equivalent distribution of multivariate convolution is also given in Omey[18](2006),and the sealing of its convolution is studied.The purpose of this thesis is to give a new definition of the equivalent distribution of multivariate convolution on the basis of the relevant research of the previous scholars,and to make further research on this basis.The first chapter of this paper introduces the research background of the equivalent distribution of multivariate convolution.In the second chapter of this paper,the correlation properties of convolution equivalence in one-dimensional situations and some definition theorems to be used in subsequent proofs are given.In the third chapter of this paper,a new definition of the equivalent distribution of multivariate convolution and its precise results obtained under some special conditions are given,and its sufficient condition and necessary conditions are studied,and the uniform convergence of its tail function and some proof methods are studied.The fourth chapter of this paper gives the application of the equivalent distribution ofmultivariate convolution in bankruptcy probability.