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November 2000 Weighted approximations of tail processes for $\beta$-mixing random variables
Holger Drees
Ann. Appl. Probab. 10(4): 1274-1301 (November 2000). DOI: 10.1214/aoap/1019487617


While the extreme value statistics for i.i.d data is well developed, much less is known about the asymptotic behavior of statistical procedures in the presence of dependence.We establish convergence of tail empirical processes to Gaussianlimits for $\beta$-mixing stationary time series. As a consequence, one obtains weighted approximations of the tail empirical quantile function that is based on a random sequence with marginal distribution belonging to the domain of attraction of an extreme value distribution. Moreover, the asymptotic normality is concluded for a large class of estimators of the extreme value index. These results are applied to stationary solutions of a general stochastic difference equation.


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Holger Drees. "Weighted approximations of tail processes for $\beta$-mixing random variables." Ann. Appl. Probab. 10 (4) 1274 - 1301, November 2000.


Published: November 2000
First available in Project Euclid: 22 April 2002

zbMATH: 1073.60520
MathSciNet: MR1810875
Digital Object Identifier: 10.1214/aoap/1019487617

Primary: 60F17 , 62M10
Secondary: 60G70 , 62G20

Keywords: $\beta$-mixing , ARCH-process , dependent , extreme value index , Hill estimator , invariance principle , statistical tail functional , stochastic difference equation , tail empirical distribution function , tail empirical quantile function , time series

Rights: Copyright © 2000 Institute of Mathematical Statistics


Vol.10 • No. 4 • November 2000
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