Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-26T16:38:20.887Z Has data issue: false hasContentIssue false
  • English
  • Français

On generalising the notion of upcrossings to random fields

Published online by Cambridge University Press:  01 July 2016

Robert J. Adler
Robert J. Adler*
Affiliation:
University of New South Wales
Get access Rights & Permissions [Opens in a new window]

Abstract

The aim of the current paper is twofold. Primarily we wish to extend some of our earlier results on excursions of random fields and introduce a more powerful technique for obtaining the mean value of a certain characteristic of these excursions. Since these results can also be used to tie together the scattered results of previous authors we also include a full review of earlier work in this field.

Keywords

RANDOM FIELDSLEVEL CROSSINGSEXCURSION SETSEXCURSION CHARACTERISTICGAUSSIAN FIELDSMEAN VALUESHIGH LEVEL EXCURSIONS
Type
Research Article
Information
Advances in Applied Probability , Volume 8 , Issue 4 , December 1976 , pp. 789 - 805
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.)

References

Adler, R. J. (1975a) Excursions Above Fixed Levels by Random Fields. Ph.D. Thesis, University of New South Wales.Google Scholar
Adler, R. J. (1975b) Excursions above high levels by Gaussian random fields. Stock. Proc. Appl. To appear.Google Scholar
Adler, R. J. (1976) Excursions above a fixed level by n-dimensional random fields. J. Appl. Prob. 13, 276–289.CrossRefGoogle Scholar
Adler, R. J. and Hasofer, A. M. (1976) Level crossings for random fields. Ann. Prob. 4, 1–12.CrossRefGoogle Scholar
Apostol, T. M. (1957) Mathematical Analysis: A Modern Approach to Advanced Calculus. Addison-Wesley, Reading, Mass.Google Scholar
Belyaev, Yu. K. (1967) Bursts and shines of random fields. Dokl. Akad. Nauk SSSR 176, 495–497.Google Scholar
Belyaev, Yu. K. (1970) Distribution of the maximum of a random field and its application to reliability problems. Eng. Cybernet. 2, 269–276.Google Scholar
Belyaev, Yu. K. (1972a) (ed.) Bursts of Random Fields (Russian), Moscow University Press.Google Scholar
Belyaev, Yu. K. (1972b) Point processes and first passage problems. Proc. 6th Berkeley Symp. Math. Statist. Prob. 2, 1–17.Google Scholar
Bickel, P. and Rosenblatt, M. (1973) Two-dimensional random fields. In Multivariate Analysis III, ed. Krishnaiah, P. R. Harcourt, Brace, Jovanovich, New York.Google Scholar
Cramér, H. and Leadbetter, M. R. (1967) Stationary and Related Stochastic Processes. Wiley, New York.Google Scholar
Dudley, R. M. (1965) Gaussian processes on several parameters. Ann. Math. Statist. 36, 771–788.CrossRefGoogle Scholar
Hasofer, A. M. (1976) The mean number of maxima above high levels in Gaussian random fields. J. Appl. Prob. 13, 377–379.CrossRefGoogle Scholar
Hasofer, A. M. and Sharpe, K. (1969) The analysis of wind gusts. Australian Meteorological Magazine 17, 198–214.Google Scholar
Jadrenko, M. I. (1967) Local properties of sample functions of random fields (Russian). Visnik Kiiv. Univ. Ser. Math. Mech. 9, 103–112; Selected Trans. Math. Statist. Prob. 10 (1971), 233–245.Google Scholar
Kac, M. (1943) On the average number of real roots of a random algebraic equation. Bull. Amer. Math. Soc. 49, 314–320.CrossRefGoogle Scholar
Leadbetter, M. R. (1972) Point processes generated by level crossings. In Stochastic Point Processes: Statistical Analysis Theory, and Applications, ed. Lewis, P.A.W. Wiley, New York.Google Scholar
Lindgren, G. (1970) Some properties of a normal process near a local maximum. Ann. Math. Statist. 41, 1870–1883.CrossRefGoogle Scholar
Lindgren, G. (1972) Local maxima of Gaussian fields. Ark. Mat. 10, 195–218.CrossRefGoogle Scholar
Lindgren, G. (1974) Spectral moment estimation by means of level crossings. Biometrika 61, 401–418.CrossRefGoogle Scholar
Longuet-Higgins, M. S. (1957) The statistical analysis of a random, moving surface. Phil. Trans. R. Soc. Lond. A 249, 321–387.Google Scholar
Malevich, T. L. (1974) A limit theorem for the length of contours generated by crossings of the zero level by Gaussian fields (Russian). Teor. Veroyat. Primenen. 19, 501–513.Google Scholar
Nosko, V. P. (1969) The characteristics of excursions of Gaussian homogeneous fields above a high level. Proceedings of the USSR–Japan Symposium on Probability, Novosibirsk.Google Scholar
Nosko, V. P. (1973) On the possibility of using the Morse inequalities for the estimation of the number of excursions of a random field in a domain. Theor. Prob. Appl. 18, 821–822.Google Scholar
Pawula, R. F. (1968) A proof of Cousin's theorem concerning stationary random surfaces. IEEE Trans. Inf. Theory IT-14, 770–772.Google Scholar
Pickands, J. iii (1969) Upcrossing probabilities for stationary Gaussian processes. Trans. Amer. Math. Soc. 145, 51–74.CrossRefGoogle Scholar
Smart, D. R. (1974) Fixed Point Theorems. Cambridge University Press.Google Scholar
Swerling, P. (1962) Statistical properties of the contours of random surfaces. I.R.E. Trans. Inf. Theory IT-8, 315–321.CrossRefGoogle Scholar