Level curves crossings and applications for Gaussian models

Extremes

Volumn 13, Issue 3, 2010, Pages 315-351

  Kratz, Marie F ^a   León, José R ^b  

^a ESSEC BUSINESS SCHOOL   (France)

^b FACULTAD DE CIENCIAS   (Venezuela)

Author keywords

(Generalized) Rice formula; CLT; Co area formula; Crossings; Gaussian fields; Harmonic oscillator; Hermite polynomials; Level curve; Specular point

Indexed keywords

EID: 77954862810     PISSN: 13861999     EISSN: 1572915X     Source Type: Journal    
DOI: 10.1007/s10687-009-0090-x     Document Type: Article

References (25)

1. Aberg, S., Rychlik, I., Leadbetter, M. R.: Palm distributions of wave characteristics in encountering seas. Ann. Probab. 18, 1059-1084 (2008).
2. Arcones, M.: Limit theorems for non-linear functionals of a stationary Gaussian sequence of vectors. Ann. Probab. 22, 2242-2274 (1994).
3. Azaïs, J.-M., León, J., Ortega, J.: Geometrical characteristic of Gaussian sea waves. J. Appl. Probab. 42, 1-19 (2005).
4. Azaïs, J.-M., Wschebor, M.: On the distribution of the maximum of a Gaussian field with d parameters. Ann. Appl. Probab. 15(1A), 254-278 (2005).
5. Azaïs, J.-M., Wschebor, M.: Level Sets and Extrema of Random Processes and Fields. Wiley, New York (2009).
6. Berry, M. V., Dennis, M. R.: Phase singularities in isotropic random waves. Proc. R. Soc. Lond. A 456, 2059-2079 (2000).
7. Breuer, J., Major, P.: Central limit theorems for non-linear functionals of Gaussian fields. J. Multivar. Anal. 13, 425-444 (1983).
8. Cabaña, E.: Esperanzas de Integrales sobre Conjuntos de Nivel aleatorios. In: Actas del 2 Congreso Latinoamericano de Probabilidad y Estadistica Matemátíca, pp. 65-82. Sociedad Bernoulli sección de Latinoamerica, Caracas (1985).
9. Cabaña, E.: Affine process: a test of isotropy based on level sets. SIAM J. App. Math. 47(4), 886-891 (1987).
10. Callenbach, L., Hänggi, P., Linz, S.: Oscillatory systems driven by noise: frequency and phase synchronization. Phys. Rev. E 65, 051110-1-051110-11 (2002).
11. Cramér, H., Leadbetter, M. R.: Stationary and Related Stochastic Processes. Wiley, New York (1967).
12. Federer, H.: Geometric Measure Theory. Springer, New York (1969).
13. Hoeffding, W., Robbins, H.: The central limit theorem for dependent random variables. Duke Math. J. 15, 773-780 (1948).
14. d and renormalization. Stoch. Process. Their Appl. 56, 1-34 (1995).
15. Kratz, M.: Level crossings and other level functionals of stationary Gaussian processes. Probability Surveys 3, 230-288 (2006).
16. Kratz, M., León, J.: Hermite polynomial expansion for non-smooth functionals of stationary Gaussian processes: crossings and extremes. Stoch. Process. Their Appl. 66, 237-252 (1997).
17. Kratz, M., León, J.: Central limit theorems for the number of maxima and some estimator of the second spectral moment of a stationary Gaussian process. Applications in hydroscience. Extremes 3(1), 57-86 (2000).
18. Kratz, M., León, J.: Central limit theorems for level functionals of stationary Gaussian processes and fields. J. Theor. Probab. 14, 639-672 (2001).
19. Kratz, M., León, J.: On the second moment of the number of crossings by a stationary Gaussian process. Ann. Probab. 34(4), 1601-1607 (2006).
20. Lindgren, G.: Slepian models for the stochastic shape of individual Lagrange sea waves. Adv. Appl. Probab. 38(2), 430-450 (2006).
21. Longuet-Higgins, M. S.: Reflection and refraction at a random surface. I, II, III. J. Opt. Soc. Am. 50(9), 838-856 (1960).
22. Longuet-Higgins, M. S.: The statistical geometry of random surfaces. In: Proc. Symp. Appl. Math., vol. XIII, pp. 105-143. AMS, Providence R. I. (1962).
23. Piterbarg, V., Rychlik, O.: Central limit theorem for wave-functionals of Gaussian processes. Adv. Appl. Probab. 31(1), 158-177 (1999).
24. Rice, S. O.: Mathematical analysis of random noise. Bell Syst. Tech. J. 23, 282-332 (1944).
25. Slud, E.: MWI representation of the number of curve-crossings by a differentiable Gaussian process, with applications. Ann. Probab. 22(3), 1355-1380 (1994).