IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v14y2001i3d10.1023_a1017597107544.html
   My bibliography  Save this article

Occupation Time Fluctuations in Branching Systems

Author

Listed:
  • D. A. Dawson

    (The Fields Institute)

  • L. G. Gorostiza

    (Centro de Investigación y de Estudios Avanzados)

  • A. Wakolbinger

    (Johann Wolfgang Goethe–Universität)

Abstract

We consider particle systems in locally compact Abelian groups with particles moving according to a process with symmetric stationary independent increments and undergoing one and two levels of critical branching. We obtain long time fluctuation limits for the occupation time process of the one- and two-level systems. We give complete results for the case of finite variance branching, where the fluctuation limits are Gaussian random fields, and partial results for an example of infinite variance branching, where the fluctuation limits are stable random fields. The asymptotics of the occupation time fluctuations are determined by the Green potential operator G of the individual particle motion and its powers G 2,G 3, and by the growth as t→∞ of the operator $$G_t = \int_0^t {T_s } ds$$ and its powers, where T t is the semigroup of the motion. The results are illustrated with two examples of motions: the symmetric α-stable Lévy process in $$\mathbb{R}^d (0

Suggested Citation

  • D. A. Dawson & L. G. Gorostiza & A. Wakolbinger, 2001. "Occupation Time Fluctuations in Branching Systems," Journal of Theoretical Probability, Springer, vol. 14(3), pages 729-796, July.
    • Handle: RePEc:spr:jotpro:v:14:y:2001:i:3:d:10.1023_a:1017597107544
    DOI: 10.1023/A:1017597107544
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1023/A:1017597107544
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1023/A:1017597107544?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. Gorostiza, L. G. & Lopez-Mimbela, J. A., 1994. "An occupation time approach for convergence of measure-valued processes, and the death process of a branching system," Statistics & Probability Letters, Elsevier, vol. 21(1), pages 59-67, September.
    2. Dawson, Donald A. & Hochberg, Kenneth J. & Vinogradov, Vladimir, 1996. "High-density limits of hierarchically structured branching-diffusing populations," Stochastic Processes and their Applications, Elsevier, vol. 62(2), pages 191-222, July.
    3. Deuschel, Jean-Dominique & Wang, Kongming, 1994. "Large deviations for the occupation time functional of a Poisson system of independent Brownian particles," Stochastic Processes and their Applications, Elsevier, vol. 52(2), pages 183-209, August.
    4. Grabner, Peter J. & Woess, Wolfgang, 1997. "Functional iterations and periodic oscillations for simple random walk on the Sierpinski graph," Stochastic Processes and their Applications, Elsevier, vol. 69(1), pages 127-138, July.
    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. D. A. Dawson & L. G. Gorostiza, 2018. "Transience and Recurrence of Random Walks on Percolation Clusters in an Ultrametric Space," Journal of Theoretical Probability, Springer, vol. 31(1), pages 494-526, March.

    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. Tomasz Bojdecki & Luis G. Gorostiza & Anna Talarczyk, 2004. "Sub-fractional Brownian motion and its relation to occupation times," RePAd Working Paper Series lrsp-TRS376, Département des sciences administratives, UQO.
    2. T. Bojdecki & Luis G. Gorostiza & A. Talarczyk, 2005. "A Long Range Dependence Stable Process and an Infinite Variance Branching System," RePAd Working Paper Series lrsp-TRS425, Département des sciences administratives, UQO.
    3. T. Bojdecki & Luis G. Gorostiza & A. Talarczyk, 2004. "Functional Limit Theorems for Occupation Time Fluctuations of Branching Systems in the Cases of Large and Critical Dimensions," RePAd Working Paper Series lrsp-TRS404, Département des sciences administratives, UQO.
    4. Bojdecki, T. & Gorostiza, L.G. & Talarczyk, A., 2008. "Occupation time limits of inhomogeneous Poisson systems of independent particles," Stochastic Processes and their Applications, Elsevier, vol. 118(1), pages 28-52, January.
    5. Luis G. Gorostiza & Reyla A. Navarro & Eliane R. Rodrigues, 2004. "Some Long-Range Dependence Processes Arising from Fluctuations of Particle Systems," RePAd Working Paper Series lrsp-TRS401, Département des sciences administratives, UQO.
    6. Bojdecki, T. & Gorostiza, L.G. & Talarczyk, A., 2006. "Limit theorems for occupation time fluctuations of branching systems I: Long-range dependence," Stochastic Processes and their Applications, Elsevier, vol. 116(1), pages 1-18, January.
    7. Bojdecki, Tomasz & Gorostiza, Luis G. & Talarczyk, Anna, 2004. "Sub-fractional Brownian motion and its relation to occupation times," Statistics & Probability Letters, Elsevier, vol. 69(4), pages 405-419, October.
    8. Bojdecki, Tomasz & Talarczyk, Anna, 2012. "Particle picture interpretation of some Gaussian processes related to fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 122(5), pages 2134-2154.
    9. Zhou, Xiaowen, 2008. "A zero-one law of almost sure local extinction for (1+[beta])-super-Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 118(11), pages 1982-1996, November.
    10. T. Bojdecki & Luis G. Gorostiza & A. Talarczyk, 2004. "Functional Limit Theorems for Occupation Time Fluctuations of Branching Systems in the Case of Long-Range Dependence," RePAd Working Paper Series lrsp-TRS402, Département des sciences administratives, UQO.
    11. Bojdecki, T. & Gorostiza, L.G. & Talarczyk, A., 2006. "Limit theorems for occupation time fluctuations of branching systems II: Critical and large dimensions," Stochastic Processes and their Applications, Elsevier, vol. 116(1), pages 19-35, January.

    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:14:y:2001:i:3:d:10.1023_a:1017597107544. 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.