IDEAS home Printed from
   My bibliography  Save this article

Approximation of stopped Brownian local time by diadic crossing chains


  • Knight, Frank B.


Let B(t) be a Brownian motion on R, B(0) = 0, and for [alpha]n:= 2-n let Tn0 = 0, Tnk+1 = inf{t> Tnk:B(t)-B(Tnk) = [alpha]n}, 0 [less-than-or-equals, slant] k. Then B(Tnk):= Rn(k[alpha]2n) is the nth approximating random walk. Define Mn by TnMn = T(-1) (the passage time to -1) and let L(x) be the local time of B at T(-1). The paper is concerned with 1. (a) the conditional law of L given [sigma](Rn), and 2. (b) the estimator E(L(·)[sigma](Rn)). Let Nn(k) denote the number of upcrossings by Rn of (k[alpha]n, (k + 1)[alpha]n) by step Mn. Explicit formulae for (a) and (b) are obtained in terms of Nn. More generally, for T = TnKn, 0[less-than-or-equals, slant]Kn [set membership, variant] [sigma](Rn), let L(x) be the local time at T, and let N±n(k) be the respective numbers of upcrossings (downcrossings) by step Kn. Simple expressions for (a) and (b) are given in terms of N±n. For fixed measure [mu] on R, 2nE[[integral operator](E(L(x)[sigma](Rn)) - L(x))2[mu](dx)[sigma](Rn] is obtained, and when [mu](dx) = dx it reduces to . With T kept fixed as n --> [infinity], this converges P-a.s. to .

Suggested Citation

  • Knight, Frank B., 1997. "Approximation of stopped Brownian local time by diadic crossing chains," Stochastic Processes and their Applications, Elsevier, vol. 66(2), pages 253-270, March.
  • Handle: RePEc:eee:spapps:v:66:y:1997:i:2:p:253-270

    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.


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

    Cited by:

    1. Ohashi, Alberto & Simas, Alexandre B., 2015. "A note on the sharp Lp-convergence rate of upcrossings to the Brownian local time," Statistics & Probability Letters, Elsevier, vol. 100(C), pages 137-141.
    2. Leão, Dorival & Ohashi, Alberto, 2010. "Weak Approximations for Wiener Functionals," Insper Working Papers wpe_215, Insper Working Paper, Insper Instituto de Ensino e Pesquisa.


    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:66:y:1997:i:2:p:253-270. 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.

    We have no 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.

    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.