IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v84y1999i1p25-51.html
   My bibliography  Save this article

The quasi-stationary distribution for small random perturbations of certain one-dimensional maps

Author

Listed:
  • Ramanan, Kavita
  • Zeitouni, Ofer

Abstract

We analyse the quasi-stationary distributions of the family of Markov chains {Xn[var epsilon]},[var epsilon]>0, obtained from small non-local random perturbations of iterates of a map f : I-->I on a compact interval. The class of maps considered is slightly more general than the class of one-dimensional Axiom A maps. Under certain conditions on the dynamics, we show that as [var epsilon]-->0 the limit quasi-stationary distribution of the family of Markov chains is supported on the union of the periodic attractors of the map f. Moreover, we show that these conditions are satisfied by Markov chains obtained as perturbations of the logistic map f(x)=[mu]x(1-x) by additive Gaussian noise and also by Markov chains that model density-dependent branching processes.

Suggested Citation

  • Ramanan, Kavita & Zeitouni, Ofer, 1999. "The quasi-stationary distribution for small random perturbations of certain one-dimensional maps," Stochastic Processes and their Applications, Elsevier, vol. 84(1), pages 25-51, November.
    • Handle: RePEc:eee:spapps:v:84:y:1999:i:1:p:25-51
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304-4149(99)00044-7
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    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. Klebaner, Fima C., 1993. "Population-dependent branching processes with a threshold," Stochastic Processes and their Applications, Elsevier, vol. 46(1), pages 115-127, May.
    2. Högnäs, Göran, 1997. "On the quasi-stationary distribution of a stochastic Ricker model," Stochastic Processes and their Applications, Elsevier, vol. 70(2), pages 243-263, October.
    3. Klebaner, F. C. & Nerman, O., 1994. "Autoregressive approximation in branching processes with a threshold," Stochastic Processes and their Applications, Elsevier, vol. 51(1), pages 1-7, June.
    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. Mao, Xuerong & Marion, Glenn & Renshaw, Eric, 2002. "Environmental Brownian noise suppresses explosions in population dynamics," Stochastic Processes and their Applications, Elsevier, vol. 97(1), pages 95-110, January.
    2. Tian, Rongrong & Wei, Jinlong & Wu, Jiang-Lun, 2021. "On a generalized population dynamics equation with environmental noise," Statistics & Probability Letters, Elsevier, vol. 168(C).

    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. Klebaner, F. C. & Liptser, R., 1999. "Moderate deviations for randomly perturbed dynamical systems," Stochastic Processes and their Applications, Elsevier, vol. 80(2), pages 157-176, April.
    2. Christine Jacob, 2010. "Branching Processes: Their Role in Epidemiology," IJERPH, MDPI, vol. 7(3), pages 1-19, March.
    3. Högnäs, Göran, 1997. "On the quasi-stationary distribution of a stochastic Ricker model," Stochastic Processes and their Applications, Elsevier, vol. 70(2), pages 243-263, October.
    4. Williams, Noah, 2004. "Small noise asymptotics for a stochastic growth model," Journal of Economic Theory, Elsevier, vol. 119(2), pages 271-298, December.
    5. Aziz Khanchi, 2012. "Asymptotics of Markov Additive Chains on a Half-Plane: A Ratio Limit Theorem," Journal of Theoretical Probability, Springer, vol. 25(1), pages 62-76, March.

    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:84:y:1999:i:1:p:25-51. 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.