IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v24y2011i1d10.1007_s10959-009-0271-1.html
   My bibliography  Save this article

The Exact Hausdorff Measure of the Zero Set of Fractional Brownian Motion

Author

Listed:
  • D. Baraka

    (École Polytechnique Fédérale)

  • T. S. Mountford

    (École Polytechnique Fédérale)

Abstract

Let {X(t), t∈ℝ N } be a fractional Brownian motion in ℝ d of index H. If L(0,I) is the local time of X at 0 on the interval I⊂ℝ N , then there exists a positive finite constant c(=c(N,d,H)) such that $$m_\phi\bigl(X^{-1}(0)\cap I\bigr)=cL(0,I),$$ where $\phi(t)=t^{N-dH}(\log\log\frac{1}{t})^{dH/N}$ , and m φ (E) is the Hausdorff φ-measure of E. This refines a previous result of Xiao (Probab. Theory Relat. Fields 109: 126–197, 1997) on the relationship between the local time and the Hausdorff measure of zero set for d-dimensional fractional Brownian motion on ℝ N .

Suggested Citation

  • D. Baraka & T. S. Mountford, 2011. "The Exact Hausdorff Measure of the Zero Set of Fractional Brownian Motion," Journal of Theoretical Probability, Springer, vol. 24(1), pages 271-293, March.
    • Handle: RePEc:spr:jotpro:v:24:y:2011:i:1:d:10.1007_s10959-009-0271-1
    DOI: 10.1007/s10959-009-0271-1
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-009-0271-1
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10959-009-0271-1?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Kasahara, Y. & Kôno, N. & Ogawa, T., 1999. "On tail probability of local times of Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 82(1), pages 15-21, July.
    2. Berman, Simeon M., 1987. "Spectral conditions for local nondeterminism," Stochastic Processes and their Applications, Elsevier, vol. 27, pages 73-84.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Boufoussi, Brahim & Guerbaz, Raby, 2009. "Smoothness of Gaussian local times beyond the local nondeterminism," Stochastic Processes and their Applications, Elsevier, vol. 119(3), pages 1001-1014, March.
    2. Bo Li & Yimin Xiao & Xiaochuan Yang, 2019. "On the Favorite Points of Symmetric Lévy Processes," Journal of Theoretical Probability, Springer, vol. 32(4), pages 1943-1972, December.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jotpro:v:24:y:2011:i:1:d:10.1007_s10959-009-0271-1. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.