Some New Sufficient Conditions And Necessary Conditions For Convolution Equivalent Distribution Class

The convolution equivalence distribution in distribution theory has become one of the key concerns of the frontier scholars,and it is widely used in the fields of queuing theory,branching process and risk theory,so it has been paid more and more attention.A.Borovkov and K.A.Borovkov have given suff-icient conditions and necessary conditions for subexponential distribution clusters in asymptotic analysis of long-tailed random variables,and given new semi-exponential distributed clusters.It is well known that the subexponential distribution S=S(0)(?=0),clusters have new suff-icient conditions and necessary conditions due to the existence of this distribution cluster.It is a special case of convolution equivalent distribution S(?)(?? 0),and there are still many basic problems in the field of convolution equivalent distribution S(?),such as in the upcoming Annals of Pro.In bability,Toshiro Watanabe gave a negative answer to the closed conjecture of convolution roots proposed by Embrechts and Goldie in 1982,and gave two sufficient conditions on the closure of convolution roots.But the conjecture that Serguei Foss and Dmitry Korshunov put forward in Annals of Probability in 2007 about the necessary and sufficient condition of convolution equivalent distribution cluster is still not solved.Based on the new mathematical tools of the 2008 A.a.borovkov and K.a.borovkov,some new sufficient conditions and necessary condition for the convolution equivalence distribution of this paper are obtained.The first chapter is an introduction,which introduces the historical background and the latest progress of the sub-exponential distribution and convolution equivalent distribution,and briefly introduces some existing achievements and the writing motivation of this paper.The second chapter introduces the basic concepts and conclusions,mainly introduces some common distribution families and the concepts of two distribution families that will be used in this paper.And the slow variable function and regular function used later in this paper.In the third chapter,we give some new necessary conditions for convolution equivalent distributive class,and prove the Lundberg exponent and simply explain some main new necessary conditions.In the fourth chapter,some new sufficient conditions for convolution equivalent distribution class,some properties of semi-exponential distribution class and their relations with convolution equivalent distribution class are discussed.