IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v26y2013i1d10.1007_s10959-011-0380-5.html
   My bibliography  Save this article

Small Deviations for a Family of Smooth Gaussian Processes

Author

Listed:
  • Frank Aurzada

    (Technische Universität Berlin)

  • Fuchang Gao

    (University of Idaho)

  • Thomas Kühn

    (Universität Leipzig)

  • Wenbo V. Li

    (University of Delaware)

  • Qi-Man Shao

    (Hong Kong University of Science and Technology)

Abstract

We study the small deviation probabilities of a family of very smooth self-similar Gaussian processes. The canonical process from the family has the same scaling property as standard Brownian motion and plays an important role in the study of zeros of random polynomials. Our estimates are based on the entropy method, discovered in Kuelbs and Li (J. Funct. Anal. 116:133–157, 1993) and developed further in Li and Linde (Ann. Probab. 27:1556–1578, 1999), Gao (Bull. Lond. Math. Soc. 36:460–468, 2004), and Aurzada et al. (Teor. Veroâtn. Ee Primen. 53:788–798, 2009). While there are several ways to obtain the result with respect to the L 2-norm, the main contribution of this paper concerns the result with respect to the supremum norm. In this connection, we develop a tool that allows translating upper estimates for the entropy of an operator mapping into L 2[0,1] by those of the operator mapping into C[0,1], if the image of the operator is in fact a Hölder space. The results are further applied to the entropy of function classes, generalizing results of Gao et al. (Proc. Am. Math. Soc. 138:4331–4344, 2010).

Suggested Citation

  • Frank Aurzada & Fuchang Gao & Thomas Kühn & Wenbo V. Li & Qi-Man Shao, 2013. "Small Deviations for a Family of Smooth Gaussian Processes," Journal of Theoretical Probability, Springer, vol. 26(1), pages 153-168, March.
    • Handle: RePEc:spr:jotpro:v:26:y:2013:i:1:d:10.1007_s10959-011-0380-5
    DOI: 10.1007/s10959-011-0380-5
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-011-0380-5
    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/s10959-011-0380-5?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. Nazarov, Alexander, 2009. "Log-level comparison principle for small ball probabilities," Statistics & Probability Letters, Elsevier, vol. 79(4), pages 481-486, February.
    2. A. A. Borovkov & P. S. Ruzankin, 2008. "On Small Deviations of Series of Weighted Random Variables," Journal of Theoretical Probability, Springer, vol. 21(3), pages 628-649, September.
    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. Seok Young Hong & Oliver Linton, 2016. "Asymptotic properties of a Nadaraya-Watson type estimator for regression functions of in finite order," CeMMAP working papers 53/16, Institute for Fiscal Studies.

    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:26:y:2013:i:1:d:10.1007_s10959-011-0380-5. 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.