IDEAS home Printed from
   My bibliography  Save this article

Phases in the two-color tenable zero-balanced Pólya process


  • Sparks, Joshua
  • Mahmoud, Hosam M.


The Pólya process is obtained by embedding the usual (discrete-time) Pólya urn scheme in continuous time. We study the class of tenable Pólya processes of white and blue balls with zero balance (no change in n, the total number of balls, over time). This class includes the (continuous-time) Ehrenfest process and the (continuous-time) Coupon Collector’s process. We look at the composition of the urn at time tn (dependent on n). We identify a critical phase of tn at the edges of which phase transitions occur. In the subcritical phase, under proper scaling the number of white balls is concentrated around a constant. In the critical phase, we have sufficient variability for an asymptotic normal distribution to be in effect. In this phase, the influence of the initial conditions is still somewhat pronounced. Beyond the critical phase, the urn is very well mixed with an asymptotic normal distribution, in which all initial conditions wither away. The results are obtained by an analytic approach utilizing partial differential equations.

Suggested Citation

  • Sparks, Joshua & Mahmoud, Hosam M., 2013. "Phases in the two-color tenable zero-balanced Pólya process," Statistics & Probability Letters, Elsevier, vol. 83(1), pages 265-271.
  • Handle: RePEc:eee:stapro:v:83:y:2013:i:1:p:265-271
    DOI: 10.1016/j.spl.2012.08.020

    Download full text from publisher

    File URL:
    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

    1. Kholfi, Sanaa & Mahmoud, Hosam M., 2012. "The class of tenable zero-balanced Pólya urns with an initially dominant subset of colors," Statistics & Probability Letters, Elsevier, vol. 82(1), pages 49-57.
    2. Srinivasan Balaji & Hosam Mahmoud, 2006. "Exact and limiting distributions in diagonal Pólya processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 58(4), pages 813-813, December.
    3. Balaji, Srinivasan & Mahmoud, Hosam M. & Watanabe, Osamu, 2006. "Distributions in the Ehrenfest process," Statistics & Probability Letters, Elsevier, vol. 76(7), pages 666-674, April.
    4. Srinivasan Balaji & Hosam Mahmoud, 2006. "Exact and Limiting Distributions in Diagonal Pólya Processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 58(1), pages 171-185, March.
    Full references (including those not matched with items on IDEAS)


    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:83:y:2013:i:1:p:265-271. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Dana Niculescu). General contact details of provider: .

    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 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.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.