IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v124y2014i1p505-521.html
   My bibliography  Save this article

A strong law of large numbers for super-stable processes

Author

Listed:
  • Kouritzin, Michael A.
  • Ren, Yan-Xia

Abstract

Let ℓ be Lebesgue measure and X=(Xt,t≥0;Pμ) be a supercritical, super-stable process corresponding to the operator −(−Δ)α/2u+βu−ηu2 on Rd with constants β,η>0 and α∈(0,2]. Put Wˆt(θ)=e(|θ|α−β)tXt(e−iθ⋅), which for each smallθ is an a.s. convergent complex-valued martingale with limit Wˆ(θ) say. We establish for any starting finite measure μ satisfying ∫Rd|x|μ(dx)<∞ that td/αXteβt→cαWˆ(0)ℓPμ-a.s. in a topology, termed the shallow topology, strictly stronger than the vague topology yet weaker than the weak topology, where cα>0 is a known constant. This result can be thought of as an extension to a class of superprocesses of Watanabe’s strong law of large numbers for branching Markov processes.

Suggested Citation

  • Kouritzin, Michael A. & Ren, Yan-Xia, 2014. "A strong law of large numbers for super-stable processes," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 505-521.
  • Handle: RePEc:eee:spapps:v:124:y:2014:i:1:p:505-521
    DOI: 10.1016/j.spa.2013.08.009
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414913002366
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.spa.2013.08.009?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.

    References listed on IDEAS

    as
    1. Blount, Douglas & Kouritzin, Michael A., 2010. "On convergence determining and separating classes of functions," Stochastic Processes and their Applications, Elsevier, vol. 120(10), pages 1898-1907, September.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Kouritzin, Michael A. & Lê, Khoa & Sezer, Deniz, 2019. "Laws of large numbers for supercritical branching Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3463-3498.
    2. Liu, Rongli & Ren, Yan-Xia & Song, Renming, 2022. "Convergence rate for a class of supercritical superprocesses," Stochastic Processes and their Applications, Elsevier, vol. 154(C), pages 286-327.
    3. Palau, Sandra & Yang, Ting, 2020. "Law of large numbers for supercritical superprocesses with non-local branching," Stochastic Processes and their Applications, Elsevier, vol. 130(2), pages 1074-1102.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Iro Ren'e Kouarfate & Michael A. Kouritzin & Anne MacKay, 2020. "Explicit solution simulation method for the 3/2 model," Papers 2009.09058, arXiv.org, revised Jan 2021.
    2. Kouritzin, Michael A. & Lê, Khoa & Sezer, Deniz, 2019. "Laws of large numbers for supercritical branching Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3463-3498.
    3. Li, Liping & Li, Xiaodan, 2020. "Dirichlet forms and polymer models based on stable processes," Stochastic Processes and their Applications, Elsevier, vol. 130(10), pages 5940-5972.

    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:124:y:2014:i:1:p:505-521. 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.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with 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.