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Central limit theorem for wave-functionals of Gaussian processes

Part of: Special processes Limit theorems Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Vladimir Piterbarg  and
Igor Rychlik
Vladimir Piterbarg*
Affiliation:
Moscow University
Igor Rychlik*
Affiliation:
University of Lund
*
∗ Postal address: Faculty of Mechanics and Mathematics, Moscow University, Vorobyovy Gory, Moscow 119 899, Russia.
∗∗ Postal address: Centre for Mathematical Sciences, University of Lund, Box 118, S-22100 Lund, Sweden. Email address: igor@maths.lth.se
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Abstract

In this paper a central limit theorem is proved for wave-functionals defined as the sums of wave amplitudes observed in sample paths of stationary continuously differentiable Gaussian processes. Examples illustrating this theory are given.

Keywords

CrossingsSlepian modelsstrong mixing propertycentral limit theorem

MSC classification

Primary: 60G15: Gaussian processes 60G35: Signal detection and filtering
Secondary: 60K40: Other physical applications of random processes 60F05: Central limit and other weak theorems
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 31 , Issue 1 , March 1999 , pp. 158 - 177
Copyright
Copyright © Applied Probability Trust 1999 

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Footnotes

Supported by Swedish Research Council for Engineering Sciences grant 908911 TFR 91-747.

Supported in part by Office of Naval Research under Grant N00014-93-1-0841.

References

Berman, S. M. (1992). Sojourns and Extremes of Stochastic Processes. Wadsworth and Brooks Cole, Andover, UK.CrossRefGoogle 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
Cuzick, J. (1976). A central limit theorem for the number of zeros of a stationary Gaussian process. Ann. Prob. 4, 547–556.CrossRefGoogle Scholar
Geman, D. (1972). On the variance of the number of zeros of a stationary Gaussian process. Ann. Math. Statist. 43, 977–982.Google Scholar
Ibragimov, I. A. and Linnik, Yu. V. (1971). Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen.Google Scholar
Ibragimov, I. A. and Rosanov, Y.A. (1978). Gaussian Random Processes. Springer, New York.Google Scholar
Leonov, V. P. (1961). On the dispersion of time-dependent means of a stationary stochastic process. Theory Prob. Appl., 6, 87–93.CrossRefGoogle Scholar
Lindgren, G. (1977). Functional limits of empirical distributions in crossing theory. Stoch. Proc. Appl. 5, 143–149.CrossRefGoogle Scholar
Lindgren, G. and Rychlik, I. (1991). Slepian models and regression approximations in crossing and extreme value theory. Internat. Statist. Rev., 59, 195–225.Google Scholar
Piterbarg, V. I. (1978). The central limit theorem for the number of level crossings of a stationary Gaussian process. Theory Prob. Appl. 23, 178–182.Google Scholar
Piterbarg, V. I. (1996). Asymptotic Methods in the Theory of Gaussian Proceses and Fields. (Transl. Math. Monogr. 148.) Amer. Math. Soc., Providence, RI.Google Scholar
Rejman, A. and Rychlik, I. (1993). Fatigue life distributions with linear and nonlinear damage rules. Statist. Research Report No. 1993:3, Univ. of Lund, pp. 1–26.Google Scholar
Rychlik, I. (1987). A note on Durbin's formula for the first passage density. Statist. Prob. Lett. 5, 425–428.Google Scholar
Rychlik, I. (1996). Extremes, rainflow cycles and damage functionals in continuous random processes. Stoch. Proc. Appl. 63, 97–116.CrossRefGoogle Scholar
Rychlik, I. Johannesson, P. and Leadbetter, M. R. (1997). Modelling and statistical analysis of ocean-wave data using transformed Gaussian processes. Marine Structures 10, 13–47.Google Scholar
Slepian, D. (1963). On the zeros of Gaussian noise. In Time Series Analysis, ed. Rosenblatt, M. Wiley, New York, 104–115.Google Scholar