IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v17y1984i2p359-367.html
   My bibliography  Save this article

Necessary and sufficient conditions for the Robbins-Monro method

Author

Listed:
  • Clark, Dean S.

Abstract

This paper examines the relation between convergence of the Robbins-Monro iterates Xn+1= Xn-an[latin small letter f with hook](Xn)+an[xi]n, [latin small letter f with hook]([theta])=0, and the laws of large numbers Sn=an[Sigma]n-1j=0 [xi]j-->0 as n-->+[infinity]. If an is decreasing at least as rapidly as c/n, then Xn-->[theta] w.p. 1 (resp. in Lp, p[greater-or-equal, slanted]1) implies Sn-->0 w.p. 1 (resp. in Lp, p[greater-or-equal, slanted]1) as n-->+[infinity]. If an is decreasing at least as slowly as c[+45 degree rule]n and limn-->+[infinity]a n=0, then Sn-->0 w.p. 1 (resp. in Lp, p[greater-or-equal, slanted]2) implies Xn-->[theta] w.p. 1 (resp. in Lp, p[greater-or-equal, slanted]2) as n -->+[infinity]. Thus, there is equivalence in the frequently examined case an[reverse similar, equals]c[+45 degree rule]n. Counter examples show that the LLN must have the form of Sn, that the rate of decrease conditions are sharp, that the weak LLN is neither necessary nor sufficient for the convergence in probability of Xn to [theta] when an[reverse similar, equals]c[+45 degree rule]n.

Suggested Citation

  • Clark, Dean S., 1984. "Necessary and sufficient conditions for the Robbins-Monro method," Stochastic Processes and their Applications, Elsevier, vol. 17(2), pages 359-367, July.
    • Handle: RePEc:eee:spapps:v:17:y:1984:i:2:p:359-367
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/0304-4149(84)90011-5
    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. Arsham H., 1998. "Techniques for Monte Carlo Optimizing," Monte Carlo Methods and Applications, De Gruyter, vol. 4(3), pages 181-230, December.
    2. Arsham Hossein, 2007. "Monte Carlo Techniques for Parametric Finite Multidimensional Integral Equations," Monte Carlo Methods and Applications, De Gruyter, vol. 13(3), pages 173-195, August.

    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:spapps:v:17:y:1984:i:2:p:359-367. 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/505572/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.