Some Equivalent Conditions Of Extended Long-tailed Distribution And Locally Long-tailed Distributions

It is well known that the distribution theory is one foundations of probability,and it has very important applications in random walk,and so in risk theory,queueing theory,Levy.process theory,branching process theory and so on,so it has always been paid wide attentions.While the extended long-tailed distribution class is one of the very broad distribution classes,it involves the convolution equivalent distribution class which has important application value,however one critical problem of the extended long-tailed distribution theory and extended convolution equivalence distribution theory is so-called the closure of convolution.In the first chapter of this article, we introduce some conventional distribution classes and local distribution classes, then recall some existing results,finally lead to the motive of this article.In the first part of the second chapter,we obtain a equivalent condition of the convolution closure of the extended long-tailed distributions,generalize and improve the corresponding results of Pakes(2004),and then receive a kind of equivalent conditions of convolution-equivalence distributions.In the last part of the second chapter,we prove these results.In the third chapter,we discuss the corresponding questions of locally long-tailed distributions,and then gain the corresponding results of the convolution closure of locally long-tailed distribution and locally subexponential distributions.