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

Let F be a proper distribution on D=[0,∞) or (−∞,∞) and N be a non-negative integer-valued random variable with masses pn=P(N=n),n≥0. Denote G=∑n=0∞pnF∗n. The main result of this paper is that under some suitable conditions, G belongs to the convolution equivalent distribution class if and only if F 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 Watanabe (2008). Similarly, some corresponding results for the local distributions and densities have been obtained.