Asymptotic expansions of convolutions of regularly varying distributions

In this paper we derive precise tail-area approximations for the sum of an arbitrary finite number of independent heavy-tailed random variables. In order to achieve second-order asymptotics, a mild regularity condition is imposed on the class of distribution functions with regularly varying tails. Higher-order asymptotics are also obtained when considering a semiparametric subclass of distribution functions with regularly varying tails. These semiparametric subclasses are shown to be closed under convolutions and a convolution algebra is constructed to evaluate the parameters of a convolution from the parameters of the constituent distributions in the convolution. A Maple code is presented which does this task.