IDEAS home Printed from https://ideas.repec.org/p/cdl/ucsdec/qt4z4380t7.html
   My bibliography  Save this paper

Asymptotic Properties of Some Projection-based Robbins-Monro Procedures in a Hilbert Space

Author

Listed:
  • Chen, Xiaohong
  • White, Halbert

Abstract

Let H be an infinite-dimentional real separable Hilbert space. Given an unknown mapping M : H (arrow) H that can only be observed with noise, we consider two modified Robbins-Monro procedures to estimate the zero point (theta) (subscript 0) ? H of M. These procedures work in appropriate finite dimensional sub-spaces of growing dimension. Almost-sure convergence, functional central limit theorem (hence asymptotic normality), law of iterated logarithm (hence almost-sure loglog rate of convergence), and mean rate of convergence are obtained for Hilbert space-valued mixingale, (theta)-dependent error processes.

Suggested Citation

  • Chen, Xiaohong & White, Halbert, 2002. "Asymptotic Properties of Some Projection-based Robbins-Monro Procedures in a Hilbert Space," University of California at San Diego, Economics Working Paper Series qt4z4380t7, Department of Economics, UC San Diego.
  • Handle: RePEc:cdl:ucsdec:qt4z4380t7
    as

    Download full text from publisher

    File URL: http://www.escholarship.org/uc/item/4z4380t7.pdf;origin=repeccitec
    Download Restriction: no

    Other versions of this item:

    References listed on IDEAS

    as
    1. Woodford, Michael, 1990. "Learning to Believe in Sunspots," Econometrica, Econometric Society, vol. 58(2), pages 277-307, March.
    2. Crawford, Vincent P, 1995. "Adaptive Dynamics in Coordination Games," Econometrica, Econometric Society, vol. 63(1), pages 103-143, January.
    3. Marcet, Albert & Sargent, Thomas J., 1989. "Convergence of least squares learning mechanisms in self-referential linear stochastic models," Journal of Economic Theory, Elsevier, vol. 48(2), pages 337-368, August.
    4. Spear, Stephen E, 1989. "Learning Rational Expectations under Computability Constraints," Econometrica, Econometric Society, vol. 57(4), pages 889-910, July.
    5. Evans, George W & Honkapohja, Seppo, 1995. "Local Convergence of Recursive Learning to Steady States and Cycles in Stochastic Nonlinear Models," Econometrica, Econometric Society, vol. 63(1), pages 195-206, January.
    6. Pakes, Ariel & McGuire, Paul, 2001. "Stochastic Algorithms, Symmetric Markov Perfect Equilibrium, and the 'Curse' of Dimensionality," Econometrica, Econometric Society, vol. 69(5), pages 1261-1281, September.
    7. Kuan, Chung-Ming & White, Halbert, 1994. "Adaptive Learning with Nonlinear Dynamics Driven by Dependent Processes," Econometrica, Econometric Society, vol. 62(5), pages 1087-1114, September.
    8. Yin, G. & Zhu, Y. M., 1990. "On H-valued Robbins-Monro processes," Journal of Multivariate Analysis, Elsevier, vol. 34(1), pages 116-140, July.
    9. Chen, Xiaohong & White, Halbert, 1998. "Central Limit And Functional Central Limit Theorems For Hilbert-Valued Dependent Heterogeneous Arrays With Applications," Econometric Theory, Cambridge University Press, vol. 14(02), pages 260-284, April.
    10. Shwartz, Adam & Berman, Nadav, 1989. "Abstract stochastic approximations and applications," Stochastic Processes and their Applications, Elsevier, vol. 31(1), pages 133-149, March.
    11. Chen, Xiaohong & White, Halbert, 1998. "Nonparametric Adaptive Learning with Feedback," Journal of Economic Theory, Elsevier, vol. 82(1), pages 190-222, September.
    12. Chen, Xiaohong & White, Halbert, 1996. "Laws of Large Numbers for Hilbert Space-Valued Mixingales with Applications," Econometric Theory, Cambridge University Press, vol. 12(02), pages 284-304, June.
    13. Nixdorf, Rainer, 1984. "An invariance principle for a finite dimensional stochastic approximation method in a Hilbert space," Journal of Multivariate Analysis, Elsevier, vol. 15(2), pages 252-260, October.
    Full references (including those not matched with items on IDEAS)

    Citations

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


    Cited by:

    1. Felipe Perez-Marti, 2000. "Private Experience in Adaptive Learning Models," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 3(2), pages 283-310, April.

    More about this item

    Keywords

    Hilbert Space.;

    Statistics

    Access and download statistics

    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:cdl:ucsdec:qt4z4380t7. 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: (Lisa Schiff). General contact details of provider: http://edirc.repec.org/data/deucsus.html .

    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.