IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v34y2021i1d10.1007_s10959-020-01009-w.html
   My bibliography  Save this article

Maximum of Catalytic Branching Random Walk with Regularly Varying Tails

Author

Listed:
  • Ekaterina Vl. Bulinskaya

    (Novosibirsk State University)

Abstract

For a continuous-time catalytic branching random walk (CBRW) on $${\mathbb {Z}}$$ Z , with an arbitrary finite number of catalysts, we study the asymptotic behavior of position of the rightmost particle when time tends to infinity. The mild requirements include regular variation of the jump distribution tail for underlying random walk and the well-known $$L\log L$$ L log L condition for the offspring numbers. In our classification, given in Bulinskaya (Theory Probab Appl 59(4):545–566, 2015), the analysis refers to supercritical CBRW. The principal result demonstrates that, after a proper normalization, the maximum of CBRW converges in distribution to a non-trivial law. An explicit formula is provided for this normalization, and nonlinear integral equations are obtained to determine the limiting distribution function. The novelty consists in establishing the weak convergence for CBRW with “heavy” tails, in contrast to the known behavior in case of “light” tails of the random walk jumps. The new tools such as “many-to-few lemma” and spinal decomposition appear ineffective here. The approach developed in this paper combines the techniques of renewal theory, Laplace transform, nonlinear integral equations and large deviations theory for random sums of random variables.

Suggested Citation

  • Ekaterina Vl. Bulinskaya, 2021. "Maximum of Catalytic Branching Random Walk with Regularly Varying Tails," Journal of Theoretical Probability, Springer, vol. 34(1), pages 141-161, March.
    • Handle: RePEc:spr:jotpro:v:34:y:2021:i:1:d:10.1007_s10959-020-01009-w
    DOI: 10.1007/s10959-020-01009-w
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-020-01009-w
    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/s10959-020-01009-w?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. Ekaterina Vladimirovna Bulinskaya, 2014. "Local Particles Numbers in Critical Branching Random Walk," Journal of Theoretical Probability, Springer, vol. 27(3), pages 878-898, September.
    2. Bhattacharya, Ayan & Hazra, Rajat Subhra & Roy, Parthanil, 2018. "Branching random walks, stable point processes and regular variation," Stochastic Processes and their Applications, Elsevier, vol. 128(1), pages 182-210.
    3. E. Yarovaya, 2017. "Positive Discrete Spectrum of the Evolutionary Operator of Supercritical Branching Walks with Heavy Tails," Methodology and Computing in Applied Probability, Springer, vol. 19(4), pages 1151-1167, December.
    Full references (including those not matched with items on IDEAS)

    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. Ray, Souvik & Hazra, Rajat Subhra & Roy, Parthanil & Soulier, Philippe, 2023. "Branching random walk with infinite progeny mean: A tale of two tails," Stochastic Processes and their Applications, Elsevier, vol. 160(C), pages 120-160.

    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:34:y:2021:i:1:d:10.1007_s10959-020-01009-w. 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: 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.