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Model fields in crossing theory: a weak convergence perspective

Published online by Cambridge University Press:  01 July 2016

Richard J. Wilson
Richard J. Wilson*
Affiliation:
University of Queensland
*
∗Postal address: Department of Mathematics, University of Queensland, St. Lucia, QLD 4067, Australia.
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Abstract

In this paper, the behavior of a Gaussian random field near an ‘upcrossing' of a fixed level is investigated by strengthening the results of Wilson and Adler (1982) to full weak convergence in the space of functions which have continuous derivatives up to order 2. In Section 1, weak convergence and model processes are briefly discussed. The model field of Wilson and Adler (1982) is constructed in Section 2 using full weak convergence. Some of its properties are also investigated. Section 3 contains asymptotic results for the model field, including the asymptotic distributions of the Lebesgue measure of a particular excursion set and the maximum of the model field as the level becomes arbitrarily high.

Keywords

TIGHTNESSSMOOTH FUNCTIONSHOMOGENEOUSGAUSSIAN RANDOM FIELDUPCROSSINGHORIZONTAL WINDOW CONDITIONINGPALM DISTRIBUTIONERGODICEXCURSION SETMAXIMUM
Type
Research Article
Information
Advances in Applied Probability , Volume 20 , Issue 4 , December 1988 , pp. 756 - 774
Copyright
Copyright © Applied Probability Trust 1988 

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Footnotes

Research supported in part by AFOSR Grant No. F49620 85 C 0144 while the author was visiting the Center for Stochastic Processes, University of North Carolina, Chapel Hill.

References

Adler, R. J. (1978) Distribution results for the occupation measures of continuous Gaussian fields. Stoch. Proc. Appl. 7, 299–310.Google Scholar
Adler, R. J. (1981a) The Geometry of Random Fields. Wiley, New York.Google Scholar
Alder, R. J. (1981b) Random field models in surface science. Bull. Int. Statist. Inst. 44, 669–681.Google Scholar
Aronowich, M. and Adler, R. J. (1986) Extrema and level crossings of ?2-processes. Adv. Appl. Prob. 18, 901–920.Google Scholar
Belyaev, Yu. K. (1967) Bursts and shines of random fields. Soviet Math. Dokl. 8, 1107–1109.Google Scholar
Belyaev, Yu. K. (1969) Elements of the general theory of random streams of events. Appendix to the Russian edition of Stationary and Related Stochastic Processes by Cramér, H. and Leadbetter, M. R., Izdat ‘Mir’, Moscow. (English translation by Leadbetter, M. R., University of North Carolina Statistics Mimeo Series, No. 703, 1970).Google Scholar
Belyaev, Yu. K. (1972) Point processes and first passage problems. Proc. Sixth Berkeley Symp. Math. Statist. Prob. 3, 1–17, University of California Press, Berkeley.Google Scholar
Berman, S. M. (1972) A class of limiting distributions of high level excursions of Gaussian processes.. Z. Wahrscheinlichkeitsth. 21, 121–134.Google Scholar
Berman, S. M. (1974) Sojourns and extremes of Gaussian processes. Ann. Prob. 2, 999–1026 (Correction (1980) 8, 999; Correction (1984), 12, 281).Google Scholar
Berman, S. M. (1982a) Sojourns and extremes of stationary processes. Ann. Prob. 10, 1–46.CrossRefGoogle Scholar
Berman, S. M. (1982b) Sojourns and extremes of a diffusion process on a fixed interval. Adv. Appl. Prob. 14, 811–832.Google Scholar
Berman, S. M. (1983) Sojourns and extremes of Fourier sums and series with random coefficients. Stoch. Proc. Appl. 15, 213–238.Google Scholar
Berman, S. M. (1985) Limit theorems for sojourns of stochastic processes. In Probability in Banach Spaces V, Lecture Notes in Mathematics 1153, Springer-Verlag, Berlin, 40–71.Google Scholar
Berman, S. M. (1987) Extreme sojourns of Gaussian processes with a point of maximum variance. Prob. Theory Rel. Fields 74, 113–124.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Cramér, H. and Leadbetter, M. R. (1967) Stationary and Related Stochastic Processes. Wiley, New York.Google Scholar
Kac, M. and Slepian, D. (1959) Large excursions of Gaussian processes. Ann. Math. Statist. 30, 1215–1228.Google Scholar
Lamperti, J. (1965) On limit theorems for Gaussian processes. Ann. Math. Statist. 36, 304–310.Google Scholar
Leadbetter, M. R. (1972) On basic results of point process theory. Proc. 6th Berkeley Symp. Math. Statist. Prob. 3, 449–462.Google Scholar
Leadbetter, M. R., Lindgren, G. and Rootzen, H. (1983) Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, New York.Google Scholar
Lindgren, G. (1970) Some properties of a normal process near a local maximum. Ann. Math. Statist. 41, 1870–1883.Google Scholar
Lindgren, G. (1971) Extreme values of stationary normal processes. Z. Wahrscheinlichkeitsth. 17, 39–47.CrossRefGoogle Scholar
Lindgren, G. (1972a) Wave-length and amplitude in Gaussian noise. Adv. Appl. Prob. 4, 81–108.CrossRefGoogle Scholar
Lindgren, G. (1972b) Wave-length and amplitude for a stationary Gaussian process after a high maximum. Z. Wahrscheinlichkeitsth. 23, 293–326.CrossRefGoogle Scholar
Lindgren, G. (1972c) Local maxima of Gaussian fields. Ark. Math. 10, 195–218.Google Scholar
Lindgren, G. (1973) Discrete wave-analysis of continuous stochastic processes. Stoch. Proc. Appl. 1, 83–105.Google Scholar
Lindgren, G. (1975a) Prediction of catastrophes and high level crossings. Bull. Intern. Statist. Inst. 46 (2), 225–240.Google Scholar
Lindgren, G. (1975b) 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
Lindgren, G. (1979) Prediction of level crossings for normal processes containing deterministic components. Adv. Appl. Prob. 11, 93–117.CrossRefGoogle Scholar
Lindgren, G. (1980) Model processes in non-linear prediction with applications to detection and alarm. Ann. Prob. 8, 775–792.Google Scholar
Lindgren, G. (1981) Jumps and bumps on random roads. J. Sound Vibration 78, 383–395.Google Scholar
Lindgren, G. (1983) Use and structure of Slepian model processes in crossing theory. In Probability and Mathematical Statistics: Essays in Honour of Carl-Gustav Esseen, ed. Gut, A. and Holst, L., Department of Mathematics, Uppsala.Google Scholar
Lindgren, G. (1984) Use and structure of Slepian model processes for prediction and detection in crossing and extreme value theory. In Statistical Extremes and Applications, ed. Tiago de Oliviera, J., Reidel, Dordrecht, 261–284.Google Scholar
Lindgren, G. (1985) Optimal prediction of level crossings in Gaussian processes and sequences. Ann. Prob. 13, 804–824.Google Scholar
de Mare, J. (1977) The behaviour of a non-differentiable stationary Gaussian process after a level crossing. Stoch. Proc. Appl. 6, 77–86.CrossRefGoogle Scholar
de Maré, J. (1980) Optimal prediction of catastrophes with application to Gaussian processes. Ann. Prob. 8, 841–850.Google Scholar
Molcanov, S. A. and Stepanov, A. K. (1979) Bursts of a Gaussian field above an elevated level. Soviet Math Dokl. 20, 1277–1281.Google Scholar
Nosko, V. P. (1969a) The characteristics of excursions of Gaussian homogeneous random fields above a high level. Proc. USSR—Japan Symp. on Prob. (Harborovsk, 1969), Novosibirsk, 216–222.Google Scholar
Nosko, V. P. (1969b) Local structure of Gaussian random fields in the vicinity of high-level light sources. Soviet Math. Dokl. 10, 1481–1484.Google Scholar
Nosko, V. P. (1986) The local structure of a homogeneous Gaussian random field in a neighborhood of high level points. Theory Prob. Appl. 30, 767–782.Google Scholar
Slepian, D. (1962) On the zeros of Gaussian noise. In Time Series Analysis, ed. Rosenblatt, M.. Wiley, New York.Google Scholar
Wilson, R. J. (1983) A Study of Model Random Fields. Thesis, University of New South Wales.Google Scholar
Wilson, R. J. (1986) Weak convergence of probability measures in spaces of smooth functions. Stoch. Proc. Appl. 23, 333–337.Google Scholar
Wilson, R. J. and Adler, R. J. (1982) The structure of Gaussian fields near a level crossing. Adv. Appl. Prob. 14, 543–565.Google Scholar