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Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)



Asymptotic analysis of a measure of variation

Authors: H. Albrecher and J. L. Teugels
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 74 (2006).
Journal: Theor. Probability and Math. Statist. 74 (2007), 1-10
MSC (2000): Primary 62G20; Secondary 62G32
Published electronically: June 25, 2007
MathSciNet review: 2336773
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Abstract: Let $ X_i$, $ i=1,\dots,n$, be a sequence of positive independent identically distributed random variables and define

$\displaystyle T_n:=\frac{X_1^2+X_2^2+\dots+X_n^2}{(X_1+X_2+\dots+X_n)^2}. $

Utilizing Karamata's theory of functions of regular variation, we determine the asymptotic behaviour of arbitrary moments $ \mathsf{E}(T_n^k)$, $ k\in\mathbb{N}$, for large $ n$, given that $ X_1$ satisfies a tail condition, akin to the domain of attraction condition from extreme value theory. As a by-product, the paper offers a new method for estimating the extreme value index of Pareto-type tails.

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Additional Information

H. Albrecher
Affiliation: Department of Mathematics, Graz University of Technology, Steyrergasse 30, A-8010 Graz, Austria

J. L. Teugels
Affiliation: Department of Mathematics, Katholieke Universiteit Leuven, W. de Croylaan 54, B-3001 Heverlee, Belgium, and EURANDOM, P.O. Box 513 - 5600 MB Eindhoven, The Netherlands

Keywords: Functions of regular variation, domain of attraction of a stable law, extreme value theory
Received by editor(s): February 1, 2005
Published electronically: June 25, 2007
Additional Notes: Supported by Fellowship F/04/009 of the Katholieke Universiteit Leuven and the Austrian Science Foundation Project S-8308-MAT
Article copyright: © Copyright 2007 American Mathematical Society

American Mathematical Society