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May 1996 The extremes of a triangular array of normal random variables
Tailen Hsing, Jürg Hüsler, Rolf-Dieter Reiss
Ann. Appl. Probab. 6(2): 671-686 (May 1996). DOI: 10.1214/aoap/1034968149


Consider a triangular array of stationary normal random variables ${\xi_{n, i}, i \geq 0, n \geq 1)$ such that ${\xi_{n, i}, i \geq 0}$ is a stationary normal sequence for each $n \geq 1$. Let $\rho_{n, j} = \corr (\xi_{n, i}, \xi_{n, i + j})$. We show that if $(1 - \rho_{n,j}) \log n \to \delta_j \epsilon (0, \infty)$ as $n \to \infty$ for some j, then the locations where the extreme values occur cluster, and if $\rho_{n,j}$ tends to 0 fast enough as $j \to \infty$ for fixed n, then ${\xi_{n, i}, i \geq 0}$ satisfies a certain weak dependence condition. Under the two conditions, it is possible to speak about an index which measures the degree of clustering. In practice, this viewpoint can provide a better approximation of the distributions of the maxima of weakly dependent normal random variables than what is directly guided by the asymptotic theory of Berman.


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Tailen Hsing. Jürg Hüsler. Rolf-Dieter Reiss. "The extremes of a triangular array of normal random variables." Ann. Appl. Probab. 6 (2) 671 - 686, May 1996.


Published: May 1996
First available in Project Euclid: 18 October 2002

zbMATH: 0855.60019
MathSciNet: MR1398063
Digital Object Identifier: 10.1214/aoap/1034968149

Primary: 60F05 , 60G10 , 60G15

Keywords: Dependence , extremal index , time series , weak convergence

Rights: Copyright © 1996 Institute of Mathematical Statistics


Vol.6 • No. 2 • May 1996
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