IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v181y2025ics0304414924002424.html
   My bibliography  Save this article

Wasserstein convergence rates for empirical measures of random subsequence of {nα}

Author

Listed:
  • Wu, Bingyao
  • Zhu, Jie-Xiang

Abstract

Fix an irrational number α. Let X1,X2,… be independent, identically distributed, integer-valued random variables with characteristic function φ, and let Sn=∑i=1nXi be the partial sums. Consider the random walk {Snα}n≥1 on the torus, where {⋅} denotes the fractional part. We study the long time asymptotic behavior of the empirical measure of this random walk to the uniform distribution under the general p-Wasserstein distance. Our results show that the Wasserstein convergence rate depends on the Diophantine properties of α and the Hölder continuity of the characteristic function φ at the origin, and there is an interesting critical phenomenon that will occur. The proof is based on the PDE approach developed by L. Ambrosio, F. Stra and D. Trevisan in Ambrosio et al. (2019) and the continued fraction representation of the irrational number α.

Suggested Citation

  • Wu, Bingyao & Zhu, Jie-Xiang, 2025. "Wasserstein convergence rates for empirical measures of random subsequence of {nα}," Stochastic Processes and their Applications, Elsevier, vol. 181(C).
    • Handle: RePEc:eee:spapps:v:181:y:2025:i:c:s0304414924002424
    DOI: 10.1016/j.spa.2024.104534
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414924002424
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2024.104534?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:eee:spapps:v:181:y:2025:i:c:s0304414924002424. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    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.