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Mesure du voisinage and occupation density
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  • Published: September 1986

Mesure du voisinage and occupation density

  • M. Csörgő1 &
  • P. Révész2 nAff3 

Probability Theory and Related Fields volume 73, pages 211–226 (1986)Cite this article

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  • 7 Citations

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Summary

Let W(t) be a standard Wiener process with occupation density (local time) η(x, t). Paul Lévy showed that for each x, η(x, t) is a.s. equal to the “mesure du voisinage” of W, i.e., to the limit as h approaches zero of h 1/2 times N(h, x, t), the number of excursions from x, exceeding h in length, that are completed by W up to time t. Recently, Edwin Perkins showed that the exceptional null sets, which may depend on x, can be combined into a single null set off which the above convergence is uniform in x. The main aim of the present paper is to estimate the rate of convergence in Perkins' theorem as h goes to zero. We also investigate the connection between N and η in the case when we observe a Wiener process through a long time t and consider the number of long (but much shorter than t) excursions.

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References

  • Csörgő, M., Révész, P.: Strong approximations in probability and statistics. Budapest: Akadémiai Kiadó and New York: Academic Press 1981

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Author information

Author notes
  1. P. Révész

    Present address: Institut für Statistik und Wahrscheinlichkeitstheorie, Technische Universität Wien, Wiedner Hauptstraße 8-10, A-1040, Wien, Austria

Authors and Affiliations

  1. Department of Mathematic and Statistics, Carleton University, K15 5B6, Ottawa

    M. Csörgő

  2. Mathematical Institute, Hungarian Academy of Sciences, Reáltanoda u. 13-15, 1053, Budapest, Hungary

    P. Révész

Authors
  1. M. Csörgő
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  2. P. Révész
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Additional information

Research partially supported by a NSERC Canada grant

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Cite this article

Csörgő, M., Révész, P. Mesure du voisinage and occupation density. Probab. Th. Rel. Fields 73, 211–226 (1986). https://doi.org/10.1007/BF00339937

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  • Received: 06 December 1984

  • Revised: 26 September 1985

  • Issue Date: September 1986

  • DOI: https://doi.org/10.1007/BF00339937

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Keywords

  • Stochastic Process
  • Probability Theory
  • Statistical Theory
  • Local Time
  • Wiener Process
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