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Approximation du temps local des surfaces gaussiennes
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  • Published: March 1993

Approximation du temps local des surfaces gaussiennes

  • Corinne Berzin1 &
  • Mario Wschebor2 

Probability Theory and Related Fields volume 96, pages 1–32 (1993)Cite this article

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  • 4 Citations

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Summary

LetX be a centered stationary Gaussian stochastic process with ad-dimensional parameter (d≧2),F its spectral measure,\(\int\limits_{R^d } {||x||^2 F(dx) = + \infty } \) (‖x‖ denotes the Euclidean norm ofx). We consider regularizations of the trajectories ofX by means of convolutions of the formX ε(t)=(Ψ ε*X)(t) where Ψε stands for an approximation of unity (as ε tends to zero) satisfying certain regularity conditions.

The aim of this paper is to recover the local time ofX at a given levelu, as a limit of appropriate normalizations of the geometric measure of theu-level set of the regular approximating processesX ε. A part of the difficulties comes from the fact that the geometric behavior of the covariance of the Gaussian processX ε can be a complex one as ε approaches O.

The results are onL 2-convergence and include bounds for the speed of convergence.L presults may be obtained in similar ways, but almost sure convergence or simultaneous convergence for the various values ofu do not seem to follow from our methods. In Sect. 3 we have included examples showing a diversity of geometric behaviors, especially in what concerns the dependence on the thickness of the set in which the covariance of the original processX is irregular.

Some technical results of analytic nature are included as appendices in Sect. 4.

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Authors and Affiliations

  1. Equipe de Statistiques, Université de Paris-Sud, Bâtiment 425, F-91405, Orsay, France

    Corinne Berzin

  2. Universidad de la República, Montevideo, Uruguay

    Mario Wschebor

Authors
  1. Corinne Berzin
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  2. Mario Wschebor
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Berzin, C., Wschebor, M. Approximation du temps local des surfaces gaussiennes. Probab. Th. Rel. Fields 96, 1–32 (1993). https://doi.org/10.1007/BF01195880

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  • Received: 27 March 1991

  • Revised: 20 September 1992

  • Issue Date: March 1993

  • DOI: https://doi.org/10.1007/BF01195880

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