IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v28y2015i4d10.1007_s10959-014-0551-2.html
   My bibliography  Save this article

Moment Asymptotics for Multitype Branching Random Walks in Random Environment

Author

Listed:
  • Onur Gün

    (Weierstrass Institute Berlin)

  • Wolfgang König

    (Weierstrass Institute Berlin
    Institute for Mathematics, TU Berlin)

  • Ozren Sekulović

    (University of Montenegro)

Abstract

We study a discrete-time multitype branching random walk on a finite space with finite set of types. Particles move in space according to a Markov chain whereas offspring distributions are given by a random field that is fixed throughout the evolution of the particles. Our main interest lies in the averaged (annealed) expectation of the population size, and its long-time asymptotics. We first derive, for fixed time, a formula for the expected population size with fixed offspring distributions, which is reminiscent of a Feynman–Kac formula. We choose Weibull-type distributions with parameter $$1/\rho _{ij}$$ 1 / ρ i j for the upper tail of the mean number of $$j$$ j type particles produced by an $$i$$ i type particle. We derive the first two terms of the long-time asymptotics, which are written as two coupled variational formulas, and interpret them in terms of the typical behavior of the system.

Suggested Citation

  • Onur Gün & Wolfgang König & Ozren Sekulović, 2015. "Moment Asymptotics for Multitype Branching Random Walks in Random Environment," Journal of Theoretical Probability, Springer, vol. 28(4), pages 1726-1742, December.
    • Handle: RePEc:spr:jotpro:v:28:y:2015:i:4:d:10.1007_s10959-014-0551-2
    DOI: 10.1007/s10959-014-0551-2
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-014-0551-2
    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/s10959-014-0551-2?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. Machado, F. P. & Popov, S. Yu., 2003. "Branching random walk in random environment on trees," Stochastic Processes and their Applications, Elsevier, vol. 106(1), pages 95-106, July.
    2. Francis Comets & Nobuo Yoshida, 2011. "Branching Random Walks in Space–Time Random Environment: Survival Probability, Global and Local Growth Rates," Journal of Theoretical Probability, Springer, vol. 24(3), pages 657-687, September.
    3. Biggins, J. D. & Cohn, H. & Nerman, O., 1999. "Multi-type branching in varying environment," Stochastic Processes and their Applications, Elsevier, vol. 83(2), pages 357-400, October.
    4. Nina Gantert & Sebastian Müller & Serguei Popov & Marina Vachkovskaia, 2010. "Survival of Branching Random Walks in Random Environment," Journal of Theoretical Probability, Springer, vol. 23(4), pages 1002-1014, December.
    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. Vladimir Kutsenko & Stanislav Molchanov & Elena Yarovaya, 2024. "Branching Random Walks in a Random Killing Environment with a Single Reproduction Source," Mathematics, MDPI, vol. 12(4), pages 1-22, February.

    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. Vincent Bansaye, 2019. "Ancestral Lineages and Limit Theorems for Branching Markov Chains in Varying Environment," Journal of Theoretical Probability, Springer, vol. 32(1), pages 249-281, March.
    2. Makoto Nakashima, 2013. "Minimal Position of Branching Random Walks in Random Environment," Journal of Theoretical Probability, Springer, vol. 26(4), pages 1181-1217, December.
    3. Vincent Bansaye & Alain Camanes, 2018. "Queueing for an infinite bus line and aging branching process," Queueing Systems: Theory and Applications, Springer, vol. 88(1), pages 99-138, February.
    4. Xiaoqiang Wang & Chunmao Huang, 2017. "Convergence of Martingale and Moderate Deviations for a Branching Random Walk with a Random Environment in Time," Journal of Theoretical Probability, Springer, vol. 30(3), pages 961-995, September.

    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:28:y:2015:i:4:d:10.1007_s10959-014-0551-2. 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.