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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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