IDEAS home Printed from https://ideas.repec.org/a/taf/lstaxx/v53y2024i15p5612-5628.html
   My bibliography  Save this article

Positive definite functions, stationary covariance functions, and Bochner’s theorem: Some results and a critical overview

Author

Listed:
  • Donato Posa

Abstract

In this article, the link among the classes of stationary covariance functions, positive definite functions, and the class of functions defined through Bochner’s theorem has been properly analyzed: indeed, in the literature, the relationship among the above classes of functions has often generated some misunderstanding. Moreover, in order to provide an exhaustive outline on the above classes, a generalization of Bochner’s theorem has been pointed out. For what concerns all the properties and results given in this article for positive definite and strictly positive definite functions, it has been underlined which of them are very general, that is, they are valid for any covariance function, hence they are independent from Bochner’s theorem and which of them are strictly related to Bochner’s representation, which provides a complete characterization for the special subclass of continuous covariance functions. On the other hand, several applications in time series, in spatial and, more generally, in spatiotemporal literature utilize covariance models which are characterized by a discontinuity at the origin, that is, a nugget effect, or are zero almost everywhere, hence these covariance functions cannot be represented through Bochner’s theorem. Some useful results on the subset of the covariance functions, which are strictly positive definite and occur in all the interpolation problems, have also been given.

Suggested Citation

  • Donato Posa, 2024. "Positive definite functions, stationary covariance functions, and Bochner’s theorem: Some results and a critical overview," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 53(15), pages 5612-5628, August.
  • Handle: RePEc:taf:lstaxx:v:53:y:2024:i:15:p:5612-5628
    DOI: 10.1080/03610926.2023.2223780
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1080/03610926.2023.2223780
    Download Restriction: Access to full text is restricted to subscribers.

    File URL: https://libkey.io/10.1080/03610926.2023.2223780?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.

    More about this item

    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:taf:lstaxx:v:53:y:2024:i:15:p:5612-5628. 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: Chris Longhurst (email available below). General contact details of provider: http://www.tandfonline.com/lsta .

    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.