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Asymptotic expansions of convolutions of regularly varying distributions

Part of: Stochastic processes Distribution theory

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: [email protected]
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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

References

[1]Beirlant, J., Teugels, J. L. and Vynckier, P., Practical analysis of extreme values (Leuven University Press, Leuven, 1996).Google Scholar
[2]Bingham, N. H., Goldie, C. M. and Teugels, J. L., Regular variation (Cambridge University Press, Cambridge, 1987).CrossRefGoogle Scholar
[3]Borovkov, A. A. and Borovkov, K. A., ‘On probabilities of large deviations for random walks. I. Regularly varying distribution tails’, Theory Probab. Appl. 46 (2001), 193–213.CrossRefGoogle Scholar
[4]Box, G. E. P., ‘Robustness is the strategy of scientific model building’, in: Statistics (eds. Launer, R. L. and Wilkinson, G. N.) (Academic Press, 1979).Google Scholar
[5]Cohen, J. W., ‘On the tail of the staitonary waiting time distribution and limit theorems for the M/G/1 queue’, Ann. Inst. H. Poincaré Probab. Statist. 8 (1972), 255–263.Google Scholar
[6]Embrechts, P. and Goldie, C. M., ‘On convolution tails’, Stochastic Process. Appl. 13 (1982), 263–278.CrossRefGoogle Scholar
[7]Embrechts, P., Klüppelberg, C. and Mikosch, T., Modeling extremal events (Springer, New York, 1997).CrossRefGoogle Scholar
[8]Feller, W., An introduction to probability theory and its applications (Wiley, New York, 1971).Google Scholar
[9]Field, C. and Ronchetti, E., Small sample asymptotics, Lecture Notes Monograph Series 13 (Insitute of Mathematical Statistics, Haywad, Califronia, 1990).CrossRefGoogle Scholar
[10]Geluk, J., Peng, L. and de Vries, G., ‘Convolutions of heavy tailed random variables and applications to diversification and MA(1) time series’, Adv. Appl. Probab. 32 (2000), 1011–1026.CrossRefGoogle Scholar
[11]Hall, P. and Weissman, I., ‘On the estimation of extreme tail probablities’, Ann. Statist. 25 (1997), 1311–1326.CrossRefGoogle Scholar
[12]Resnick, S. I., ‘Point processes, regular variation and weak convergence’, Adv. Appl. Probab. 18 (1986), 66–138.CrossRefGoogle Scholar
[13]Resnick, S. I., Extreme values, regular variation, and point processes (Springer, New York, 1987).CrossRefGoogle Scholar
[14]Willmot, G. E. and Lin, X. S., Lundberg approximations for compound distributions with insurances applications (Springer, New York, 2001).CrossRefGoogle Scholar