IDEAS home Printed from https://ideas.repec.org/a/eee/stapro/v8y1989i2p189-192.html
   My bibliography  Save this article

A proof of the Markov chain tree theorem

Author

Listed:
  • Anantharam, V.
  • Tsoucas, P.

Abstract

Let X be a finite set, P be a stochastic matrix on X, and = limn --> [infinity] (1/n)[summation operator]n-1k=0Pk. Let G = (X, E) be the weighted directed graph on X associated to P, with weights pij. An arborescence is a subset a [subset, double equals] E which has at most one edge out of every node, contains no cycles, and has maximum possible cardinality. The weight of an arborescence is the product of its edge weights. Let denote the set of all arborescences. Let ij denote the set of all arborescences which have j as a root and in which there is a directed path from i to j. Let [short parallel][short parallel], resp. [short parallel]ij[short parallel], be the sum of the weights of the arborescences in , resp. ij. The Markov chain tree theorem states that ij = [short parallel]ij[short parallel]/[short parallel][short parallel]. We give a proof of this theorem which is probabilistic in nature.

Suggested Citation

  • Anantharam, V. & Tsoucas, P., 1989. "A proof of the Markov chain tree theorem," Statistics & Probability Letters, Elsevier, vol. 8(2), pages 189-192, June.
    • Handle: RePEc:eee:stapro:v:8:y:1989:i:2:p:189-192
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/0167-7152(89)90016-3
    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.

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Dubey, Pradeep & Sahi, Siddhartha & Shubik, Martin, 2018. "Money as minimal complexity," Games and Economic Behavior, Elsevier, vol. 108(C), pages 432-451.
    2. L. Avena & A. Gaudillière, 2018. "Two Applications of Random Spanning Forests," Journal of Theoretical Probability, Springer, vol. 31(4), pages 1975-2004, December.
    3. Candogan, Ozan & Ozdaglar, Asuman & Parrilo, Pablo A., 2013. "Dynamics in near-potential games," Games and Economic Behavior, Elsevier, vol. 82(C), pages 66-90.
    4. Choi, Michael C.H. & Huang, Zhipeng, 2023. "Generalized Markov chain tree theorem and Kemeny’s constant for a class of non-Markovian matrices," Statistics & Probability Letters, Elsevier, vol. 193(C).

    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:8:y:1989:i:2:p:189-192. 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.

    We have no bibliographic references for this item. You can help adding them by using 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.