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December, 1975 Weak Convergence of High Level Crossings and Maxima for One or More Gaussian Processes
Georg Lindgren, Jacques de Mare, Holger Rootzen
Ann. Probab. 3(6): 961-978 (December, 1975). DOI: 10.1214/aop/1176996222

Abstract

Weak convergence of the multivariate point process of upcrossings of several high levels by a stationary Gaussian process is established. The limit is a certain multivariate Poisson process. This result is then used to determine the joint asymptotic distribution of heights and locations of the highest local maxima over an increasing interval. The results are generalized to upcrossings and local maxima of two dependent Gaussian processes. To prevent nuisance jitter from hiding the overall structure of crossings and maxima the above results are phrased in terms of $\varepsilon$-crossings and $\varepsilon$-maxima, but it is shown that under suitable regularity conditions the results also hold for ordinary upcrossings and maxima.

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Georg Lindgren. Jacques de Mare. Holger Rootzen. "Weak Convergence of High Level Crossings and Maxima for One or More Gaussian Processes." Ann. Probab. 3 (6) 961 - 978, December, 1975. https://doi.org/10.1214/aop/1176996222

Information

Published: December, 1975
First available in Project Euclid: 19 April 2007

zbMATH: 0328.60021
MathSciNet: MR385997
Digital Object Identifier: 10.1214/aop/1176996222

Subjects:
Primary: 60G15
Secondary: 60F05 , 60G10 , 60G17

Keywords: dependent processes , local maxima , Stationary Gaussian processes , upcrossings , weak convergence

Rights: Copyright © 1975 Institute of Mathematical Statistics

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Vol.3 • No. 6 • December, 1975
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