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Asymptotic Expansions for Distributions of Compound Sums of Random Variables with Rapidly Varying Subexponential Distribution

Part of: Special processes Limit theorems Stochastic processes Approximations and expansions

Published online by Cambridge University Press:  14 July 2016

Ph. Barbe ,
W. P. McCormick  and
C. Zhang
Ph. Barbe*
Affiliation:
CNRS
W. P. McCormick*
Affiliation:
University of Georgia
C. Zhang*
Affiliation:
University of Georgia
*
∗Postal address: 90 rue de Vaugirard, 75006 Paris, France.
∗∗Postal address: Department of Statistics, University of Georgia, Athens, GA 30602, USA.
∗∗Postal address: Department of Statistics, University of Georgia, Athens, GA 30602, USA.
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Abstract

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We derive an asymptotic expansion for the distribution of a compound sum of independent random variables, all having the same rapidly varying subexponential distribution. The examples of a Poisson and geometric number of summands serve as an illustration of the main result. Complete calculations are done for a Weibull distribution, with which we derive, as examples and without any difficulties, seven-term expansions.

Keywords

Asymptotic expansionconvolutiontail area approximationregular variationsubexponential distribution

MSC classification

Secondary: 60F99: None of the above, but in this section 60K05: Renewal theory 60G50: Sums of independent random variables; random walks 41A60: Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
Type
Research Article
Information
Journal of Applied Probability , Volume 44 , Issue 3 , September 2007 , pp. 670 - 684
Copyright
Copyright © Applied Probability Trust 2007 

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