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Theory of Probability and Mathematical Statistics

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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
DOI: https://doi.org/10.1090/S0094-9000-07-00692-8
Published electronically: June 25, 2007
MathSciNet review: 2336773
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Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. H. Albrecher, S. Ladoucette, and J. Teugels, Asymptotics of the Sample Coefficient of Variation and the Same Dispersion, K. U. Leuven UCS Report 2006-04, 2006.
  • 2. J. Beirlant, Y. Goegebeur, J. Segers, and J. Teugels, Statistics of Extremes: Theory and Applications, Wiley, Chichester, 2004. MR 2108013 (2005j:62002)
  • 3. N. Bingham, C. Goldie, and J. Teugels, Regular variation, Encyclopedia of Mathematics and its Applications, vol. 27, Cambridge University Press, Cambridge, 1987. MR 898871 (88i:26004)
  • 4. H. Cohn and P. Hall, On the limit behaviour of weighted sums of random variables, Z. Wahrsch. Verw. Gebiete 59 (1982), no. 3, 319-331. MR 721629 (85g:60029)
  • 5. A. Fuchs, A. Joffe, and J. Teugels, Expectation of the ratio of the sum of squares to the square of the sum: exact and asymptotic results, Theory Probab. Appl. 46 (2001), no. 2, 243-255. MR 1968687 (2004b:62045)
  • 6. D. L. McLeish and G. L. O'Brien, The expected ratio of the sum of squares to the square of the sum, Ann. Probab. 10 (1982), no. 4, 1019-1028. MR 672301 (84a:60039)

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Additional Information

H. Albrecher
Affiliation: Department of Mathematics, Graz University of Technology, Steyrergasse 30, A-8010 Graz, Austria
Email: albrecher@TUGraz.at

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
Email: jef.teugels@wis.kuleuven.ac.be

DOI: https://doi.org/10.1090/S0094-9000-07-00692-8
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

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