IDEAS home Printed from https://ideas.repec.org/a/eee/stapro/v50y2000i3p293-304.html
   My bibliography  Save this article

Random motions, classes of ergodic Markov chains and beta distributions

Author

Listed:
  • Stoyanov, Jordan
  • Pirinsky, Christo

Abstract

We consider classes of discrete time Markov chains with continuous state space, the interval (0,1). These chains arise as stochastic models of phenomena in areas such as population theory, motion of particles in a random environment, etc. We exploit the Fréchet-Shohat theorem to establish that these Markov chains are ergodic and find explicitly their ergodic distributions as being beta distributions. Then we show that the convergence in total variation norm is at a geometric rate. Related topics are also discussed.

Suggested Citation

  • Stoyanov, Jordan & Pirinsky, Christo, 2000. "Random motions, classes of ergodic Markov chains and beta distributions," Statistics & Probability Letters, Elsevier, vol. 50(3), pages 293-304, November.
    • Handle: RePEc:eee:stapro:v:50:y:2000:i:3:p:293-304
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167-7152(00)00114-0
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    References listed on IDEAS

    as
    1. Devroye, Luc & Letac, Gerard & Seshadri, Vanamamalai, 1986. "The limit behavior of an interval splitting scheme," Statistics & Probability Letters, Elsevier, vol. 4(4), pages 183-186, June.
    2. Robert B. Lund & Richard L. Tweedie, 1996. "Geometric Convergence Rates for Stochastically Ordered Markov Chains," Mathematics of Operations Research, INFORMS, vol. 21(1), pages 182-194, February.
    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. Ladjimi, Fetima & Peigné, Marc, 2019. "On the asymptotic behavior of the Diaconis–Freedman chain on [0,1]," Statistics & Probability Letters, Elsevier, vol. 145(C), pages 1-11.
    2. McKinlay, Shaun, 2017. "On beta distributed limits of iterated linear random functions," Statistics & Probability Letters, Elsevier, vol. 127(C), pages 33-41.
    3. Pacheco-González, Carlos G., 2009. "Ergodicity of a bounded Markov chain with attractiveness towards the centre," Statistics & Probability Letters, Elsevier, vol. 79(20), pages 2177-2181, October.
    4. Letac, Gérard, 2002. "Donkey walk and Dirichlet distributions," Statistics & Probability Letters, Elsevier, vol. 57(1), pages 17-22, March.
    5. Liu, Yujie & Niu, Minwen & Yao, Dacheng & Zhang, Hanqin, 2022. "Stationary distributions and ergodicity of reflection-type Markov chains," Statistics & Probability Letters, Elsevier, vol. 189(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. Hervé, Loïc & Ledoux, James, 2014. "Approximating Markov chains and V-geometric ergodicity via weak perturbation theory," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 613-638.
    2. McKinlay, Shaun, 2017. "On beta distributed limits of iterated linear random functions," Statistics & Probability Letters, Elsevier, vol. 127(C), pages 33-41.
    3. Kam C. Yuen & Yuhua Lu & Rong Wu, 2009. "The compound Poisson process perturbed by a diffusion with a threshold dividend strategy," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 25(1), pages 73-93, January.
    4. Gareth O. Roberts & Jeffrey S. Rosenthal, 2019. "Hitting Time and Convergence Rate Bounds for Symmetric Langevin Diffusions," Methodology and Computing in Applied Probability, Springer, vol. 21(3), pages 921-929, September.
    5. Xi, Fubao & Yin, G., 2010. "Asymptotic properties of nonlinear autoregressive Markov processes with state-dependent switching," Journal of Multivariate Analysis, Elsevier, vol. 101(6), pages 1378-1389, July.
    6. Bar Light, 2024. "A Note on the Stability of Monotone Markov Chains," Papers 2401.11568, arXiv.org, revised Sep 2024.
    7. Daly, Fraser, 2024. "On Stein’s method for stochastically monotone single-birth chains," Statistics & Probability Letters, Elsevier, vol. 206(C).
    8. Johnson, Norman L. & Kotz, Samuel, 1995. "Use of moments in studies of limit distributions arising from iterated random subdivisions of an interval," Statistics & Probability Letters, Elsevier, vol. 24(2), pages 111-119, August.
    9. Margarete Knape & Ralph Neininger, 2008. "Approximating Perpetuities," Methodology and Computing in Applied Probability, Springer, vol. 10(4), pages 507-529, December.
    10. Wenpin Tang, 2019. "Exponential ergodicity and convergence for generalized reflected Brownian motion," Queueing Systems: Theory and Applications, Springer, vol. 92(1), pages 83-101, June.
    11. Roberts, G. O. & Tweedie, R. L., 1999. "Bounds on regeneration times and convergence rates for Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 80(2), pages 211-229, April.

    More about this item

    Keywords

    ;

    Statistics

    Access and download statistics

    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:stapro:v:50:y:2000:i:3:p:293-304. 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/622892/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.