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On the rate of convergence for extremes of mean square differentiable stationary normal processes

Part of: Stochastic processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Marie F. Kratz  and
Holger Rootzén
Marie F. Kratz*
Affiliation:
Université René Descartes
Holger Rootzén*
Affiliation:
Chalmers University of Technology
*
∗Postal address: UFR de Mathématiques et Informatique, Université René Descartes, Paris V, 45 rue des Saints-Pères, 75270 Paris Cedex 06, France. e-mail: kratz@math-info.univ-paris5.fr
∗∗Postal address: Department of Mathematics, Chalmers University of Technology, S-41962 Göteborg, Sweden. e-mail: rootzen@math.chalmers.se
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Abstract

Let ξ (t); t ≧ 0 be a normalized continuous mean square differentiable stationary normal process with covariance function r(t). Further, let and set . We give bounds which are roughly of order Τ –δ for the rate of convergence of the distribution of the maximum and of the number of upcrossings of a high level by ξ (t) in the interval [0, T]. The results assume that r(t) and r′(t) decay polynomially at infinity and that r″ (t) is suitably bounded. For the number of upcrossings it is in addition assumed that r(t) is non-negative.

Keywords

RATE OF CONVERGENCEEXTREMESNORMAL PROCESSESPOISSON CONVERGENCE

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60G15: Gaussian processes 60F05: Central limit and other weak theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 34 , Issue 4 , December 1997 , pp. 908 - 923
Copyright
Copyright © Applied Probability Trust 1997 

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