IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v25y2012i3d10.1007_s10959-010-0333-4.html
   My bibliography  Save this article

Haar-Based Multiresolution Stochastic Processes

Author

Listed:
  • Wei Zhang

    (University of Houston)

  • Marjorie G. Hahn

    (Tufts University)

Abstract

Modifying a Haar wavelet representation of Brownian motion yields a class of Haar-based multiresolution stochastic processes in the form of an infinite series $$X_t = \sum_{n=0}^\infty\lambda_n\varDelta _n(t)\epsilon_n,$$ where λ n Δ n (t) is the integral of the nth Haar wavelet from 0 to t, and ε n are i.i.d. random variables with mean 0 and variance 1. Two sufficient conditions are provided for X t to converge uniformly with probability one. Each stochastic process , the collection of all almost sure uniform limits, retains the second-moment properties and the same roughness of sample paths as Brownian motion, yet lacks some of the features of Brownian motion, e.g., does not have independent and/or stationary increments, is not Gaussian, is not self-similar, or is not a martingale. Two important tools are developed to analyze elements of , the nth-level self-similarity of the associated bridges and the tree structure of dyadic increments. These tools are essential in establishing sample path results such as Hölder continuity and fractional dimensions of graphs of the processes.

Suggested Citation

  • Wei Zhang & Marjorie G. Hahn, 2012. "Haar-Based Multiresolution Stochastic Processes," Journal of Theoretical Probability, Springer, vol. 25(3), pages 890-909, September.
    • Handle: RePEc:spr:jotpro:v:25:y:2012:i:3:d:10.1007_s10959-010-0333-4
    DOI: 10.1007/s10959-010-0333-4
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-010-0333-4
    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/s10959-010-0333-4?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.

    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:25:y:2012:i:3:d:10.1007_s10959-010-0333-4. 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.

    We have no bibliographic references for this item. You can help adding them by using 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.