Asymptotic analysis of a measure of variation
Authors:
H. Albrecher and J. L. Teugels
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: Let $X_i$, $i=1,\dots ,n$, be a sequence of positive independent identically distributed random variables and define \[ T_n:=\frac {X_1^2+X_2^2+\dots +X_n^2}{(X_1+X_2+\dots +X_n)^2}. \] 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
- 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.
- Jan Beirlant, Yuri Goegebeur, Jozef Teugels, and Johan Segers, Statistics of extremes, Wiley Series in Probability and Statistics, John Wiley & Sons, Ltd., Chichester, 2004. Theory and applications; With contributions from Daniel De Waal and Chris Ferro. MR 2108013
- N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular variation, Encyclopedia of Mathematics and its Applications, vol. 27, Cambridge University Press, Cambridge, 1987. MR 898871
- Harry Cohn and Peter Hall, On the limit behaviour of weighted sums of random variables, Z. Wahrsch. Verw. Gebiete 59 (1982), no. 3, 319–331. MR 721629, DOI https://doi.org/10.1007/BF00532224
- A. Fuks, A. Ioffe, and Dzh. Toĭgel′s, Expectation of the ratio of the sum of squares to the square of the sum: exact and asymptotic results, Teor. Veroyatnost. i Primenen. 46 (2001), no. 2, 297–310 (Russian, with Russian summary); English transl., Theory Probab. Appl. 46 (2003), no. 2, 243–255. MR 1968687, DOI https://doi.org/10.1137/S0040585X97978919
- 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
References
- 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.
- J. Beirlant, Y. Goegebeur, J. Segers, and J. Teugels, Statistics of Extremes: Theory and Applications, Wiley, Chichester, 2004. MR 2108013 (2005j:62002)
- 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)
- 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)
- 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)
- 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
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