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The rate of convergence of extremes of stationary normal sequences

Published online by Cambridge University Press:  01 July 2016

Holger Rootzén
Holger Rootzén*
Affiliation:
University of Copenhagen
*
∗Postal address; Institute of Mathematical Statistics, 5 Universitetsparken, University of Copenhagen, DK-2100 Copenhagen 0, Denmark.
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Abstract

Let {ξ; t = 1, 2, …} be a stationary normal sequence with zero means, unit variances, and covariances let be independent and standard normal, and write . In this paper we find bounds on which are roughly of the order where ρ is the maximal correlation, ρ =sup {0, r1, r2, …}. It is further shown that, at least for m-dependent sequences, the bounds are of the right order and, in a simple example, the errors are evaluated numerically. Bounds of the same order on the rate of convergence of the point processes of exceedances of one or several levels are obtained using a ‘representation' approach (which seems to be of rather wide applicability). As corollaries we obtain rates of convergence of several functionals of the point processes, including the joint distribution function of the k largest values amongst ξ1, …, ξn.

Keywords

MAXIMA OF NORMAL SEQUENCESEXTREME VALUES
Type
Research Article
Information
Advances in Applied Probability , Volume 15 , Issue 1 , March 1983 , pp. 54 - 80
Copyright
Copyright © Applied Probability Trust 1983 

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Footnotes

Research supported in part by the Office of Naval Research under contract N 0014-75-C-0809.

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