The Closure Of Some Distributions Under Convolutions And Under Convolution Roots And Their Applications

It is well known that the convolution equivalent distributions have wide and important applications in risk theory, queueing system, branching process and infinite devisibility and so on. So they have been paid wide attentions. While the closure of these distrbutions under convoltions and under convolution roots is one of the most important problems. Thhis paper obtains the closure of the convolution equivalent distributions under convolution roots. On the basis of this result, we give the asymptotics of the tails of random sums,and show that the corresponding results of Pakes(2004) is still right,although the proof there is wrong. Moreover, we obtain corresponding results for distribution densities and local distributions. Above results generalize the corresponding results of Wang et al(2006). In addition, we obtain equivalent conditions for the local closure and local asymptotics of non-identent distributions under convolution, and obtain the local asymptotics of the symmetrized random variables; these results generalize the corresponding results of Embrechts and Goldie(1980) and Gleuk(2004).