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Distributions that are both subexponential and in the domain of attraction of an extreme-value distribution

Published online by Cambridge University Press:  01 July 2016

Charles M. Goldie  and
Sidney Resnick
Charles M. Goldie*
Affiliation:
University of Sussex
Sidney Resnick*
Affiliation:
Colorado State University
*
∗Postal address: Statistical Laboratory, 16 Mill Lane, Cambridge CB2 1SB, UK.
∗∗Postal address: Department of Operations Research, Upson Hall, Cornell University, Ithaca, NY 14853, USA.
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Abstract

When does a distribution F have the property of both being in the domain of attraction of exp {–e–x} and having a second convolution-power tail equivalent to the first: Sufficient conditions and examples are given.

Keywords

SUBEXPONENTIAL DISTRIBUTIONSEXTREME-VALUE DISTRIBUTIONSCONVOLUTIONS
Type
Research Article
Information
Advances in Applied Probability , Volume 20 , Issue 4 , December 1988 , pp. 706 - 718
Copyright
Copyright © Applied Probability Trust 1988 

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Footnotes

Supported during 1985–1986 by a UK Science and Engineering Research Council Grant and at the end by NSF Grant MCS 8501763. The hospitality of the Mathematics Division, University of Sussex, is acknowledged.

References

Balkema, A. A. and Haan, L. De (1972) On R. von Mises' condition for the domain of attraction of exp{–e–x}. Ann. Math. Statist. 43, 1352–1354.Google Scholar
Balkema, A. A. and Haan, L. De (1974) Residual life-time at great age. Ann. Prob. 2, 792–804.Google Scholar
Beard, R. E., Pentikäinen, T. and Pesonen, E. (1984) Risk Theory. Chapman & Hall, London.Google Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987) Regular Variation. Encyclopedia of Mathematics and its Applications Vol. 27, Cambridge University Press.Google Scholar
Boos, D. (1987) Detecting skewed errors from regression residuals. Technometrics 29, 83–90.CrossRefGoogle Scholar
Chistyakov, V. (1964) A theorem on sums of independent, positive random variables and its applications to branching processes. Theory Prob. Appl. 9, 640–648.Google Scholar
Chover, J., Ney, P. and Wainger, S. (1973) Functions of probability measures. J. d'Anal. Math. 26, 255–302.Google Scholar
Cline, D. (1983) Estimation and Linear Prediction for Regression, Autoregression and ARMA with Infinite-variance Data. , Department of Statistics, Colorado State University, Fort Collins, CO 80523, USA.Google Scholar
Cline, D. (1986) Convolution tails, product tails and domains of attraction. Prob. Theory Rel. Fields 72, 529–557.CrossRefGoogle Scholar
Cline, D. (1987) Convolutions of distributions with exponential and subexponential tails. J. Austral. Math. Soc. A 43, 347–365.Google Scholar
Davis, R. A. and Resnick, S. I. (1985a) Limit theory for moving averages of random variables with regularly varying tail probabilities. Ann. Prob. 13, 179–195.Google Scholar
Davis, R. A. and Resnick, S. I. (1985b) More limit theory for the sample correlation function of moving averages. Stoch. Proc. Appl. 20, 257–279.Google Scholar
Davis, R. A. and Resnick, S. I. (1986) Limit theory for sample covariance and correlation functions of moving averages. Ann. Statist. 14, 533–588.Google Scholar
Embrechts, P. (1985) Subexponential distribution functions and their applications: a review. Proc. Seventh Conf. Probability Theory, Bra?o?, Romania, 1982, VNU Science Press, Utrecht, 125–136.Google Scholar
Embrechts, P. and Goldie, C. M. (1980) On closure and factorization properties of subexponential and related distributions. J. Austral. Math. Soc. A 29, 243–256.Google Scholar
Embrechts, P., Goldie, C. M. and Veraverbeke, N. (1979) Subexponentiality and infinite divisibility. Z. Warhscheinlichkeitsth. 49, 335–347.CrossRefGoogle Scholar
Ettinger, P. (1963) Etude du domaine d'attraction de la somme de deux variables aléatoires indépendantes quand l'une est du domaine d'attraction de f1 , a. C. R. Acad. Sci. Paris 257, 3810–3813.Google Scholar
Ettinger, P. (1966) Sur l'attraction du produit de deux variables aléatoires positives indépendantes par f1, a quand l'un des facteurs est attiré par f1, a. C. R. Acad. Sci. Paris 262, A696–A699.Google Scholar
Ettinger, P. (1976) Thèse, Université de Toulouse.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 2, 2nd edn. Wiley, New York.Google Scholar
Goldie, C. M. and Resnick, S. I. (1988) Subexponential distribution tails and point processes. Stoch Models 4 (2).Google Scholar
Haan, L. De (1970) On Regular Variation and its Application to the Weak Convergence of Sample Extremes. Mathematical Centre Tract 32, Mathematical Centre, Amsterdam.Google Scholar
Hogg, R. V. and Klugman, S. A. (1984) Loss Distributions. Wiley, New York.CrossRefGoogle Scholar
Klüppelberg, C. (1987) On an unpublished lemma of W. L. Smith. Preprint, University of Mannheim, D-6800 Mannheim, West Germany.Google Scholar
Klüppelberg, C. (1987) Subexponentielle Verteilungen und Charakterisierungen verwandter Klassen. Thesis, University of Mannheim.Google Scholar
Klüppelberg, C. (1988) Subexponential distributions and integrated tails. J. Appl. Prob. 25, 132–141.Google Scholar
Leslie, J. R. (1989) On the non-closure under convolution of the subexponential family. J. Appl. Prob. To appear.CrossRefGoogle Scholar
Pinelis, I. F. (1985) Asymptotic equivalence of the probabilities of large deviations for sums and maxima of independent random variables (in Russian). In Limit Theorems of Probability Theory, ed. Borovkov, A. A., pp. 144–173. Trudy Instituta Matematiki, Novosibirsk , Tom 5. Izdat. ‘Nauka’, Novosibirsk.Google Scholar
Pitman, E. J. G. (1980) Subexponential distribution functions. J. Austral. Math. Soc. A 29, 337–347.CrossRefGoogle Scholar
Resnick, S. I. (1986) Point processes, regular variation and weak convergence. Adv. Appl. Prob. 18, 66–138.CrossRefGoogle Scholar
Rootzen, H. (1978) Extremes of moving averages of stable processes. Ann. Prob. 6, 847–869.CrossRefGoogle Scholar
Rootzen, H. (1983) Extreme-value theory for moving-average processes. Preprint 6, Institute of Mathematical Statistics, University of Copenhagen.Google Scholar
Rootzen, H. (1986) Extreme-value theory for moving-average processes. Ann. Prob. 14, 612–652.Google Scholar
Sgibnev, M. S. (1986) Infinitely divisible distributions belonging to the class ℒ(γ). Contributed paper, 1st World Congress of the Bernoulli Soc., Tashkent, 1986; to appear, Sibirsk Mat. Zh. Google Scholar
Willekens, E. (1986) Hogere Orde Theorie voor Subexponentiële Verdelingen. Thesis, Mathematics Department, Katholieke Universiteit, Leuven, Belgium.Google Scholar