Hostname: page-component-7b7567c8c-2rsb6 Total loading time: 0 Render date: 2026-11-12T05:07:58.731Z Has data issue: false hasContentIssue false
  • English
  • Français

Excursions above a fixed level by n-dimensional random fields

Published online by Cambridge University Press:  14 July 2016

Robert J. Adler
Robert J. Adler*
Affiliation:
University of New South Wales
*
* Currently visiting Tel-Aviv University on a C.S.I.R.O. post-doctoral fellowship.
Get accessRights & Permissions [Opens in a new window]

Abstract

For an n-dimensional random field X(t) we define the excursion set A of X(t) by A = [t ∊ S: X(t) ≧ u] for real u and compact S ⊂ Rn. We obtain a generalisation of the number of upcrossings of a one-dimensional stochastic process to random fields via a characteristic of the set A related to the Euler characteristic of differential topology. When X(t) is a homogeneous Gaussian field satisfying certain regularity conditions we obtain an explicit formula for the mean value of this characteristic.

Keywords

RANDOM FIELDSEXCURSION SETSEULER CHARACTERISTICHOMOGENEOUS GAUSSIAN PROCESSMEAN VALUES

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 13 , Issue 2 , June 1976, pp. 276 - 289
Copyright
Copyright © Applied Probability Trust 1976 

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. and Hasofer, A. M. (1976) Level crossings for random fields. Ann. Prob. To appear.CrossRefGoogle Scholar
Aitken, A. C. (1946) Determinants and Matrices. Oliver and Boyd, London.Google Scholar
Belyaev, Yu. K. (1966) On the number of intersections of a level by a Gaussian stochastic process, I. Theor. Prob. Appl. 11, 106–113.Google Scholar
Belyaev, Yu. K. (1972a) Bursts of Random Fields (In Russian). Moscow Univ. Press.Google Scholar
Belyaev, Yu. K. (1972b) Point processes and first passage problems. Proc. 6th Berkeley Symp. Math. Statist. Prob. 3, 1–17.Google Scholar
Bickel, P. and Rosenblatt, M. (1973) Two-dimensional random fields, in Multivariate Analysis III , ed. Krishnaiah, P. R., Harcourt, Brace, Jovanavich, New York.Google Scholar
Cramér, H. and Leadbetter, M. R. (1965) The moments of the number of crossings of a level by a stationary normal process. Ann. Math. Statist. 36, 1656–1663.Google Scholar
Cramer, H. and Leadbetter, M. R. (1967) Stationary and Related Stochastic Processes. Wiley, New York.Google Scholar
Eilenberg, S. and Steenrod, N. (1952) Foundations of Algebraic Topology. Princeton University Press.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications , Vol. II. 2nd edn. Wiley, New York.Google Scholar
Kac, M. (1943) On the average number of real roots of a random algebraic equation. Bull. Amer. Math. Soc. 49, 314–320.Google Scholar
Lin, Y. K. (1967) Probabilistic Theory of Structural Dynamics. McGraw Hill, New York.Google Scholar
Lindgren, G. (1974) Spectral moment estimation by means of level crossings. Biometrika 61, 401–418.Google Scholar
Morse, M. and Cairns, S. (1969) Critical Point Theory in Global Analysis and Differential Topology. Academic Press, New York.Google Scholar
Wallace, A. H. (1968) Differential Topology: First Steps. W. A. Benjamin, New York.Google Scholar