IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v16y2003i1d10.1023_a1022242924198.html
   My bibliography  Save this article

On the Link Between Small Ball Probabilities and the Quantization Problem for Gaussian Measures on Banach Spaces

Author

Listed:
  • S. Dereich
  • F. Fehringer
  • A. Matoussi
  • M. Scheutzow

Abstract

Let μ be a centered Gaussian measure on a separable Banach space E and N a positive integer. We study the asymptotics as N→∞ of the quantization error, i.e., the infimum over all subsets ℰ of E of cardinality N of the average distance w.r.t. μ to the closest point in the set ℰ. We compare the quantization error with the average distance which is obtained when the set ℰ is chosen by taking N i.i.d. copies of random elements with law μ. Our approach is based on the study of the asymptotics of the measure of a small ball around 0. Under slight conditions on the regular variation of the small ball function, we get upper and lower bounds of the deterministic and random quantization error and are able to show that both are of the same order. Our conditions are typically satisfied in case the Banach space is infinite dimensional.

Suggested Citation

  • S. Dereich & F. Fehringer & A. Matoussi & M. Scheutzow, 2003. "On the Link Between Small Ball Probabilities and the Quantization Problem for Gaussian Measures on Banach Spaces," Journal of Theoretical Probability, Springer, vol. 16(1), pages 249-265, January.
    • Handle: RePEc:spr:jotpro:v:16:y:2003:i:1:d:10.1023_a:1022242924198
    DOI: 10.1023/A:1022242924198
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1023/A:1022242924198
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1023/A:1022242924198?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. Wenbo V. Li & Qi-Man Shao, 1999. "Small Ball Estimates for Gaussian Processes under Sobolev Type Norms," Journal of Theoretical Probability, Springer, vol. 12(3), pages 699-720, July.
    2. V. Li, Wenbo, 2001. "Small ball probabilities for Gaussian Markov processes under the Lp-norm," Stochastic Processes and their Applications, Elsevier, vol. 92(1), pages 87-102, March.
    3. David M. Mason & Zhan Shi, 2001. "Small Deviations for Some Multi-Parameter Gaussian Processes," Journal of Theoretical Probability, Springer, vol. 14(1), pages 213-239, January.
    4. Thomas Dunker, 2000. "Estimates for the Small Ball Probabilities of the Fractional Brownian Sheet," Journal of Theoretical Probability, Springer, vol. 13(2), pages 357-382, April.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Steffen Dereich, 2003. "Small Ball Probabilities Around Random Centers of Gaussian Measures and Applications to Quantization," Journal of Theoretical Probability, Springer, vol. 16(2), pages 427-449, April.
    2. S. Dereich & C. Vormoor, 2011. "The High Resolution Vector Quantization Problem with Orlicz Norm Distortion," Journal of Theoretical Probability, Springer, vol. 24(2), pages 517-544, June.
    3. Siegfried Graf & Harald Luschgy & Gilles Pagès, 2003. "Functional Quantization and Small Ball Probabilities for Gaussian Processes," Journal of Theoretical Probability, Springer, vol. 16(4), pages 1047-1062, October.

    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. 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.
    2. Siegfried Graf & Harald Luschgy & Gilles Pagès, 2003. "Functional Quantization and Small Ball Probabilities for Gaussian Processes," Journal of Theoretical Probability, Springer, vol. 16(4), pages 1047-1062, October.
    3. David M. Mason & Zhan Shi, 2001. "Small Deviations for Some Multi-Parameter Gaussian Processes," Journal of Theoretical Probability, Springer, vol. 14(1), pages 213-239, January.
    4. 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.

    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:16:y:2003:i:1:d:10.1023_a:1022242924198. 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.