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Limit distributions for compounded sums of extreme order statistics

Part of: Limit theorems Real functions

Published online by Cambridge University Press:  14 July 2016

Jan Beirlant  and
Jozef L. Teugels
Jan Beirlant
Affiliation:
Katholieke Universiteit Leuven
Jozef L. Teugels*
Affiliation:
Katholieke Universiteit Leuven
*
∗Postal address for both authors: K. U. Leuven, Departement Wiskunde, Celestijnenlaan 200B, B-3030 Leuven (Heverlee), Belgium.
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Abstract

Let X(1) ≦ X(2) ≦ ·· ·≦ X(N(t)) be the order statistics of the first N(t) elements from a sequence of independent identically distributed random variables, where {N(t); t ≧ 0} is a renewal counting process independent of the sequence of X's. We give a complete description of the asymptotic distribution of sums made from the top kt extreme values, for any sequence kt such that kt → ∞, kt/t → 0 as t → ∞. We discuss applications to reinsurance policies based on large claims.

Keywords

EXTREME VALUESASYMPTOTICSSTABLE LAWSREGULAR VARIATIONRENEWAL COUNTING PROCESSES

MSC classification

Primary: 26A12: Rate of growth of functions, orders of infinity, slowly varying functions
Secondary: 60F05: Central limit and other weak theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 29 , Issue 3 , September 1992 , pp. 557 - 574
Copyright
Copyright © Applied Probability Trust 1992 

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References

Beirlant, J. and Teugels, J. L. (1987) Asymptotics of Hill's estimator. Theory Prob. Appl. 31, 463–469.Google Scholar
Beirlant, J. and Teugels, J. L. (1989) Asymptotic normality of Hill's estimator. In Extreme Value Theory, Proceedings, Oberwolfach 1987, pp. 148–155. Lecture Notes in Statistics 51, Springer-Verlag, Berlin.Google Scholar
Beirlant, J. and Teugels, J. L. (1991) Asymptotics for compounded sums of extremes. Preprint Series, Dept. Wiskunde, Katholieke Universiteit Leuven 3, nr. 3.Google Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987) Regular Variation. Encyclopedia of Mathematics and its Applications 27, Cambridge University Press.Google Scholar
Csörgő, S. and Mason, D. M. (1985) Central limit theorems for sums of extreme values. Math. Proc. Camb. Phil. Soc. 98, 547–588.Google Scholar
Csörgő, S. and Mason, D. M. (1986) The asymptotic distribution of sums of extreme values from a regularly varying distribution. Ann Prob. 14, 974–983.Google Scholar
Embrechts, P. and Veraverbeke, N. (1982) Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance: Mathematics and Economics 1, 55–72.Google Scholar
de Haan, L. (1975) On Regular Variation and its Application to the Weak Convergence of Sample Extremes. Mathematical Centre Tract 32, Mathematisch Centrum, Amsterdam.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. II. Wiley, New York.Google Scholar
Goldie, C. M. and Smith, R. L. (1987) Slow variation with remainder: theory and applications. Quart. J. Math. Oxford 38, 45–71.CrossRefGoogle Scholar
Goovaerts, M. J., Kaas, R., van Heerwaerden, A. E. and Bauwelinckx, T. (1990) Effective Actuarial Methods. Insurance Series, Vol. 3, North-Holland, Amsterdam.Google Scholar
Haeusler, E. and Teugels, J. L. (1985) On asymptotic normality of Hill's estimator for the exponent of regular variation. Ann. Statist. 13, 743–756.CrossRefGoogle Scholar
Hall, P. (1982) On simple estimates of an exponent of regular variation. J. R. Statist. Soc. B44, 37–42.Google Scholar
Hill, B. M. (1975) A simple general approach to inference about the tail of a distribution. Ann. Statist. 3, 1163–1174.Google Scholar
Lô, G. S. (1987) Asymptotic behavior of Hill's estimate and applications. J. Appl. Prob. 23, 922–936.Google Scholar
Lô, G. S. (1989) A note on the asymptotic normality of sums of extreme values. J. Statist. Planning Inf. 22, 127–136.Google Scholar
Mason, D. M. (1982) Laws of large numbers for sums of extreme values. Ann. Prob. 10, 754–764.Google Scholar
Omey, E. and Willekens, E. (1988) ?-variation with remainder. J. London Math. Soc. 2, 105–118.Google Scholar
Teugels, J. L. (1981) Limit theorems on order statistics. Ann. Prob. 9, 868–880.Google Scholar
Teugels, J. L. (1985) Selected Topics in Insurance Mathematics. Lecture Notes Katholieke Universiteit Leuven, Dept. Wiskunde.Google Scholar
Thépaut, A. (1950) Une nouvelle forme de reassurance. Le traité d'excédent du coût moyen relatif (Ecomor). Bull. Trimestr. Inst. Actuaires Franç. 49, 273.Google Scholar