The closure of the convolution equivalent distribution class under convolution roots with applications to random sums

Abstract: Let be a proper distribution on or and be a non-negative integer-valued random variable with masses . Denote . The main result of this paper is that under some suitable conditions, belongs to the convolution equivalent distribution class if and only if belongs to the convolution equivalent distribution class. As applications, some known results on random sums have been extended and improved, which give a positive answer under certain conditions to Problem 1 of . Similarly, some corresponding results for the local distributions and densities have been obtained. [Copyright &y& Elsevier]