Closure of some heavy-tailed distribution classes under random convolution*

In this paper, we consider the closure property of a random convolution $ sumnolimits_{n = 0}^infty {{p_n}{F^*}^n} $ , where F is a heavy-tailed distribution on [0, ??), and p _n (n?=?0, 1, . . . ) are the local probabilities of a nonnegative integer-valued random variable. We obtain conditions under which the fact that distribution F belongs to the dominatedly varying-tailed class, long-tailed class, or to the intersection of these classes implies that $ sumnolimits_{n = 0}^infty {{p_n}{F^*}^n} $ is in the same class.