Hostname: page-component-7b7567c8c-tnzcf Total loading time: 0 Render date: 2026-11-12T22:26:09.482Z Has data issue: false hasContentIssue false
  • English
  • Français

The structure of Gaussian fields near a level crossing

Published online by Cambridge University Press:  01 July 2016

Richard J. Wilson and
Robert J. Adler
Richard J. Wilson*
Affiliation:
University of New South Wales
Robert J. Adler*
Affiliation:
Technion-Israel Institute of Technology
*
∗Postal address: Department of Statistics, School of Mathematics, University of New South Wales, P.O. Box 1, Kensington, NSW 2033, Australia.
∗∗Postal address: Faculty of Industrial Engineering and Management, Technion-Israel Institute of Technology, Haifa 32000, Israel.
Get accessRights & Permissions [Opens in a new window]

Abstract

In this paper, we investigate the behaviour of a Gaussian random field after an ‘upcrossing' of a particular level. In Section 1, we briefly discuss model processes and their background, and give a definition of an upcrossing of a level for random fields. A model field is constructed for the random field after an upcrossing of a level by using horizontal window conditioning in Section 2. The final section contains asymptotic distributions for the model field and for the location and height of the ‘closest' maximum to the upcrossing as the level becomes arbitrarily high.

Keywords

HOMOGENEOUSGAUSSIAN RANDOM FIELDUPCROSSINGHORIZONTAL WINDOW CONDITIONINGMODEL FIELD

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 14 , Issue 3 , September 1982, pp. 543 - 565
Copyright
Copyright © Applied Probability Trust 1982 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Adler, R. J. (1976) On generalizing the notion of upcrossings to random fields. Adv. Appl. Prob. 8, 789–805.Google Scholar
Adler, R. J. (1977) Excursions above high levels by Gaussian random fields. Stoch. Proc. Appl. 5, 21–25.Google Scholar
Adler, R. J. (1981) The Geometry of Random Fields. Wiley, London.Google Scholar
Apostol, T. M. (1957) Mathematical Analysis. Addison-Wesley, Reading, MA.Google Scholar
Belyaev, Yu. K. (1967) Bursts and shines of random fields. Soviet Math. Dokl. 8, 1107–1109.Google Scholar
Belyaev, Yu. K. and Nosko, V. P. (1969) Characteristics of excursions above a level for a Gaussian process and its envelope. Theory Prob. Appl. 16, 296–309.Google Scholar
Chung, K. L. (1974) A Course in Probability Theory, 2nd edn. Academic Press, New York.Google Scholar
Kac, M. and Slepian, D. (1959) Large excursions of Gaussian processes. Ann. Math. Statist. 30, 1215–1228.CrossRefGoogle Scholar
Lindgren, G. (1970) Some properties of a normal process near a local maximum. Ann. Math. Statist. 41, 1870–1883.Google Scholar
Lindgren, G. (1972) Local maxima of Gaussian fields. Ark. Math. 10, 195–218.Google Scholar
Lindgren, G. (1975) Prediction from a random time point. Ann. Prob. 3, 412–423.Google Scholar
Lindgren, G. (1977) Functional limits of empirical distributions in crossing theory. Stoch. Proc. Appl. 5, 143–149.Google Scholar
Slepian, D. (1962) On the zeros of Gaussian noise. In Time Series Analysis, ed. Rosenblatt, M. Wiley, New York.Google Scholar