The Annals of Applied Probability

Weighted approximations of tail processes for $\beta$-mixing random variables

Holger Drees

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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.

Article information

Ann. Appl. Probab., Volume 10, Number 4 (2000), 1274-1301.

First available in Project Euclid: 22 April 2002

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Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 60G70: Extreme value theory; extremal processes 62G20: Asymptotic properties

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 $\beta$-mixing


Drees, Holger. Weighted approximations of tail processes for $\beta$-mixing random variables. Ann. Appl. Probab. 10 (2000), no. 4, 1274--1301. doi:10.1214/aoap/1019487617.

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