IDEAS home Printed from https://ideas.repec.org/h/spr/sprchp/978-1-4757-2024-2_7.html
   My bibliography  Save this book chapter

Ergodic Theorems

In: Probability, Random Processes, and Ergodic Properties

Author

Listed:
  • Robert M. Gray

    (Stanford University, Department of Electrical Engineering)

Abstract

At the heart of ergodic theory are the ergodic theorems: results providing sufficient conditions for dynamical systems or random processes to possess ergodic properties, that is, for sample averages of the form $$ _n = \tfrac{1}{n}\sum\limits_{i = 0}^{n - 1} {fT^i } $$ to converge to an invariant limit. Traditional developments of the pointwise ergodic theorem focus on stationary systems and use a subsidiary result known as the maximal ergodic lemma (or theorem) to prove the ergodic theorem. The general result for AMS systems then follows since an AMS source inherits ergodic properties from its stationary mean; that is, since the set {x: n (x) converges } is invariant and since a system and its stationary mean place equal probability on all invariant sets, one will possess almost everywhere ergodic properties with respect to a class of measurements if and only if the other one does and the limiting sample averages will be the same.

Suggested Citation

  • Robert M. Gray, 1988. "Ergodic Theorems," Springer Books, in: Probability, Random Processes, and Ergodic Properties, chapter 7, pages 216-243, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4757-2024-2_7
    DOI: 10.1007/978-1-4757-2024-2_7
    as

    Download full text from publisher

    To our knowledge, this item is not available for download. To find whether it is available, there are three options:
    1. Check below whether another version of this item is available online.
    2. Check on the provider's web page whether it is in fact available.
    3. Perform a
    for a similarly titled item that would be available.

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    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:sprchp:978-1-4757-2024-2_7. 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.