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Approximation du temps local des processus gaussiens stationnaires par régularisation des trajectoires
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  • Published: September 1987

Approximation du temps local des processus gaussiens stationnaires par régularisation des trajectoires

  • J. M. Azaïs1 &
  • D. Florens-Zmirou2 

Probability Theory and Related Fields volume 76, pages 121–132 (1987)Cite this article

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Résumé

Soit X un processus gaussien stationnaire non dérivable. Nous étudions le nombre de passages en zéro du processus régularisé par convolution. Sous des hypothèses peu restrictives sur X, cette variable convenablement normalisée, converge au sens de L 2 quand la taille du filtre tend vers zéro. Lorsque X admet un temps local continu, la limite obtenue est le temps local.

Summary

Let {X(t)} be a stationary non differentiable Gaussian process and let ϕɛ(u=ɛ−1 ϕ(u/ɛ) be an approximate identity. Setting X ɛ(t)=X*ϕɛ(t) and letting N ɛ(T) be the number of zeros of X ɛ in the interval [0, T] it is shown that under weak technical conditions there are constants C(ɛ) so that C(ɛ) N ɛ(T) converges in L 2 as ɛ→0. When X admits a continuous local time, the limit is the local time L(0, T) at zero of X(t).

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Bibliographie

  1. Azais, J.M.: Approximation du temps local des mouvements stables. A paraître

  2. Bermann, S.M.: Local non determinism and local times of gaussian processes. Indiana Univ. Math. J. 23, 69–94 (1973)

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  3. Cramer, H., Leadbetter, H.R.: Stationary and related processes. New York: Wiley 1967

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  4. Kac, M.: On the average number of real roots of a random algebraic equation. Bull. Am. Math. Soc. 44, 314–320 (1943)

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  5. Wschebor, M.: Surfaces aléatoires: mesure géométrique des ensembles de niveau. Lect. Notes Math. 1147 (1985)

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Authors and Affiliations

  1. Statistiques appliquées UA 743, INRA Biométrie, Route de St. Cyr, F-78000, Versailles, France

    J. M. Azaïs

  2. Statistiques appliquées UA 743, Université Paris V, 45 Rue des St. Pères, F-75270, Paris Cédex 06, France

    D. Florens-Zmirou

Authors
  1. J. M. Azaïs
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  2. D. Florens-Zmirou
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Azaïs, J.M., Florens-Zmirou, D. Approximation du temps local des processus gaussiens stationnaires par régularisation des trajectoires. Probab. Th. Rel. Fields 76, 121–132 (1987). https://doi.org/10.1007/BF00390279

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  • Received: 18 December 1985

  • Revised: 30 March 1987

  • Issue Date: September 1987

  • DOI: https://doi.org/10.1007/BF00390279

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