J. Aust. Math. Soc.  78 (2005), 339-371

Asymptotic expansions of convolutions of regularly varying distributions

Philippe Barbe   CNRS   90 rue de Vaugirard   75006 Paris   France   and William P. McCormick   Department of Statistics   University of Georgia   Athens, GA 30602   USA   bill@stat.uga.edu

Abstract

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.

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