IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v36y2023i4d10.1007_s10959-023-01256-7.html
   My bibliography  Save this article

On the Ergodicity of Certain Markov Chains in Random Environments

Author

Listed:
  • Balázs Gerencsér

    (Alfréd Rényi Institute of Mathematics and Eötvös Loránd University)

  • Miklós Rásonyi

    (Alfréd Rényi Institute of Mathematics and Eötvös Loránd University)

Abstract

We study the ergodic behaviour of a discrete-time process X which is a Markov chain in a stationary random environment. The laws of $$X_t$$ X t are shown to converge to a limiting law in (weighted) total variation distance as $$t\rightarrow \infty $$ t → ∞ . Convergence speed is estimated, and an ergodic theorem is established for functionals of X. Our hypotheses on X combine the standard “drift” and “small set” conditions for geometrically ergodic Markov chains with conditions on the growth rate of a certain “maximal process” of the random environment. We are able to cover a wide range of models that have heretofore been intractable. In particular, our results are pertinent to difference equations modulated by a stationary (Gaussian) process. Such equations arise in applications such as discretized stochastic volatility models of mathematical finance.

Suggested Citation

  • Balázs Gerencsér & Miklós Rásonyi, 2023. "On the Ergodicity of Certain Markov Chains in Random Environments," Journal of Theoretical Probability, Springer, vol. 36(4), pages 2093-2125, December.
    • Handle: RePEc:spr:jotpro:v:36:y:2023:i:4:d:10.1007_s10959-023-01256-7
    DOI: 10.1007/s10959-023-01256-7
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-023-01256-7
    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/s10959-023-01256-7?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:36:y:2023:i:4:d:10.1007_s10959-023-01256-7. 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.