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Asymptotic expansions of convolutions of regularly varying distributions

Part of: Distribution theory Stochastic processes

Published online by Cambridge University Press:  09 April 2009

Philippe Barbe  and
William P. McCormick
Philippe Barbe
Affiliation:
CNRS 90 rue de Vaugirard 75006 Paris, France
William P. McCormick
Affiliation:
Department of StatisticsUniversity of Georgia Athens, GA 30602USA e-mail: bill@stat.uga.edu
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Abstract

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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 asemiparametric 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.

Keywords

convolutionconvolution algebratail areaasymptotic expansionregular variationheavy tail

MSC classification

Secondary: 62E17: Approximations to distributions (nonasymptotic) 62E20: Asymptotic distribution theory 60G70: Extreme value theory; extremal processes
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 78 , Issue 3 , June 2005 , pp. 339 - 371
Copyright
Copyright © Australian Mathematical Society 2005

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