IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v22y2009i4d10.1007_s10959-008-0187-1.html
   My bibliography  Save this article

An Optimal Wavelet Series Expansion of the Riemann–Liouville Process

Author

Listed:
  • Helga Schack

    (FSU Jena)

Abstract

Consider the Riemann–Liouville process R α ={R α (t)} t∈[0,1] with parameter α>1/2. Depending on α, wavelet series representations for R α (t) of the form ∑ k=1 ∞ u k (t)ε k are given, where the u k are deterministic functions, and {ε k } k≥1 is a sequence of i.i.d. standard normal random variables. The expansion is based on a modified Daubechies wavelet family, which was originally introduced in Meyer (Rev. Mat. Iberoam. 7:115–133, 1991). It is shown that these wavelet series representations are optimal in the sense of Kühn–Linde (Bernoulli 8:669–696, 2002) for all values of α>1/2.

Suggested Citation

  • Helga Schack, 2009. "An Optimal Wavelet Series Expansion of the Riemann–Liouville Process," Journal of Theoretical Probability, Springer, vol. 22(4), pages 1030-1057, December.
    • Handle: RePEc:spr:jotpro:v:22:y:2009:i:4:d:10.1007_s10959-008-0187-1
    DOI: 10.1007/s10959-008-0187-1
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-008-0187-1
    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/s10959-008-0187-1?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. Dzhaparidze, Kacha & Zanten, Harry van, 2005. "Optimality of an explicit series expansion of the fractional Brownian sheet," Statistics & Probability Letters, Elsevier, vol. 71(4), pages 295-301, March.
    2. Eduard Belinsky & Werner Linde, 2002. "Small Ball Probabilities of Fractional Brownian Sheets via Fractional Integration Operators," Journal of Theoretical Probability, Springer, vol. 15(3), pages 589-612, July.
    3. Antoine Ayache & Werner Linde, 2008. "Approximation of Gaussian Random Fields: General Results and Optimal Wavelet Representation of the Lévy Fractional Motion," Journal of Theoretical Probability, Springer, vol. 21(1), pages 69-96, March.
    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. Istas, Jacques, 2006. "Karhunen-Loeve expansion of spherical fractional Brownian motions," Statistics & Probability Letters, Elsevier, vol. 76(14), pages 1578-1583, August.
    2. Fuchang Gao & Wenbo V. Li, 2006. "Logarithmic Level Comparison for Small Deviation Probabilities," Journal of Theoretical Probability, Springer, vol. 19(3), pages 535-556, December.
    3. Antoine Ayache & Werner Linde, 2008. "Approximation of Gaussian Random Fields: General Results and Optimal Wavelet Representation of the Lévy Fractional Motion," Journal of Theoretical Probability, Springer, vol. 21(1), pages 69-96, March.
    4. Anatoliy Malyarenko, 2008. "An Optimal Series Expansion of the Multiparameter Fractional Brownian Motion," Journal of Theoretical Probability, Springer, vol. 21(2), pages 459-475, June.
    5. Palle Jorgensen & Feng Tian, 2019. "Realizations and Factorizations of Positive Definite Kernels," Journal of Theoretical Probability, Springer, vol. 32(4), pages 1925-1942, December.
    6. Alexandre Richard, 2017. "Some Singular Sample Path Properties of a Multiparameter Fractional Brownian Motion," Journal of Theoretical Probability, Springer, vol. 30(4), pages 1285-1309, December.
    7. Fuchang Gao & Wenbo V. Li, 2007. "Logarithmic Level Comparison for Small Deviation Probabilities," Journal of Theoretical Probability, Springer, vol. 20(1), pages 1-23, March.

    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:22:y:2009:i:4:d:10.1007_s10959-008-0187-1. 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.