IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v36y2023i3d10.1007_s10959-022-01227-4.html
   My bibliography  Save this article

Selfsimilar Free Additive Processes and Freely Selfdecomposable Distributions

Author

Listed:
  • Makoto Maejima

    (Keio University)

  • Noriyoshi Sakuma

    (Nagoya City University)

Abstract

In the paper by Fan (Inf Dim Anal Quant Probab Rel Topics 9:451–469, 2006), he introduced the marginal selfsimilarity of non-commutative stochastic processes and proved that the marginal distributions of selfsimilar processes with freely independent increments are freely selfdecomposable. In this paper, we firstly introduce a new definition, stronger than Fan’s in general, of selfsimilarity via linear combinations of non-commutative stochastic processes, although the two definitions are equivalent for non-commutative stochastic processes with freely independent increments. We secondly prove the converse of Fan’s result, to complete the relationship between selfsimilar free additive processes and freely selfdecomposable distributions. Furthermore, we construct stochastic integrals with respect to free additive processes for representing the background driving free Lévy processes of freely selfdecomposable distributions. A relationship between freely selfdecomposable distributions and their background driving free Lévy processes in terms of their free cumulant transforms is also given, and several examples are discussed.

Suggested Citation

  • Makoto Maejima & Noriyoshi Sakuma, 2023. "Selfsimilar Free Additive Processes and Freely Selfdecomposable Distributions," Journal of Theoretical Probability, Springer, vol. 36(3), pages 1667-1697, September.
    • Handle: RePEc:spr:jotpro:v:36:y:2023:i:3:d:10.1007_s10959-022-01227-4
    DOI: 10.1007/s10959-022-01227-4
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-022-01227-4
    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/s10959-022-01227-4?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:36:y:2023:i:3:d:10.1007_s10959-022-01227-4. 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.