Local Asymptotic Behavior For Random Sums With Family Of Generalized Local Convolution Equivalent Distributions And Their Applications

Asymptotic behavior of random sums is a classic but still vibrant area of research,and it has important applications of the risk theory, queuing system and other fields.It is known that the relation between asymptotic behavior of random sums and thedistribution theory is close.This paper introduces and discusses a new family of distributions, aiming atfurther extending the existing asymptotic results of random sums.In chapterⅡ, we introduce and discuss the above new family of distributions, wemight call it family of generalized local convolution equivalent distributions, and sys-tematically discuss its properties, in particular, local asymptotic behavior for randomsums.In chapterⅢ, we apply the above to the renewal equation, the compound Poissonprocess and infinitely divisible distributions and so on.To illustrate family of generalized local convolution equivalent distributions reallyembracing the familiar convolution equivalent distributions S(γ)(γ＞0) and localsubexponential distribution S_Δ(0＜T＜∞), we give a counterexample in chapterⅣ. It is worth mentioning that the example also negates a suspicion which is proposedby Foss and Korshunov (2007).