IDEAS home Printed from https://ideas.repec.org/a/eee/phsmap/v389y2010i20p4299-4316.html
   My bibliography  Save this article

Universal first-passage properties of discrete-time random walks and Lévy flights on a line: Statistics of the global maximum and records

Author

Listed:
  • Majumdar, Satya N.

Abstract

In these lecture notes I will discuss the universal first-passage properties of a simple correlated discrete-time sequence {x0=0,x1,x2,…,xn} up to n steps where xi represents the position at step i of a random walker hopping on a continuous line by drawing independently, at each time step, a random jump length from an arbitrary symmetric and continuous distribution (it includes, e.g., the Lévy flights). I will focus on the statistics of two extreme observables associated with the sequence: (i) its global maximum and the time step at which the maximum occurs and (ii) the number of records in the sequence and their ages. I will demonstrate how the universal statistics of these observables emerge as a consequence of Pollaczek–Spitzer formula and the associated Sparre Andersen theorem.

Suggested Citation

  • Majumdar, Satya N., 2010. "Universal first-passage properties of discrete-time random walks and Lévy flights on a line: Statistics of the global maximum and records," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(20), pages 4299-4316.
  • Handle: RePEc:eee:phsmap:v:389:y:2010:i:20:p:4299-4316
    DOI: 10.1016/j.physa.2010.01.021
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378437110000750
    Download Restriction: Full text for ScienceDirect subscribers only. Journal offers the option of making the article available online on Science direct for a fee of $3,000

    File URL: https://libkey.io/10.1016/j.physa.2010.01.021?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.

    Citations

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


    Cited by:

    1. Claude Godreche & Satya N. Majumdar & Gregory Schehr, 2015. "Record statistics for random walk bridges," Papers 1505.06053, arXiv.org, revised Jan 2016.
    2. Claude Godreche & Satya N. Majumdar & Gregory Schehr, 2017. "Record statistics of a strongly correlated time series: random walks and L\'evy flights," Papers 1702.00586, arXiv.org.

    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:phsmap:v:389:y:2010:i:20:p:4299-4316. 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.journals.elsevier.com/physica-a-statistical-mechpplications/ .

    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.