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

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  • Chen Xiaohong

    (Department of Economics, London School of Economics & Political Science)

  • White Halbert

    (Department of Economics, University of California, San Diego)

Abstract

Let H be an infinite-dimensional real separable Hilbert space. Given an unknown mapping M:H (r)H that can only be observed with noise, we consider two modified Robbins-Monro procedures to estimate the zero point ?o ( 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, (-dependent error processes.

Suggested Citation

  • Chen Xiaohong & White Halbert, 2002. "Asymptotic Properties of Some Projection-based Robbins-Monro Procedures in a Hilbert Space," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 6(1), pages 1-55, April.
    • Handle: RePEc:bpj:sndecm:v:6:y:2002:i:1:n:1
    DOI: 10.2202/1558-3708.1000
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    2. Kristensen, Dennis & Salanié, Bernard, 2017. "Higher-order properties of approximate estimators," Journal of Econometrics, Elsevier, vol. 198(2), pages 189-208.
    3. Dennis Kristensen & Bernard Salanié, 2010. "Higher Order Improvements for Approximate Estimators," CAM Working Papers 2010-04, University of Copenhagen. Department of Economics. Centre for Applied Microeconometrics.
    4. Chen, Xiaohong, 2007. "Large Sample Sieve Estimation of Semi-Nonparametric Models," Handbook of Econometrics, in: J.J. Heckman & E.E. Leamer (ed.), Handbook of Econometrics, edition 1, volume 6, chapter 76, Elsevier.
    5. Xiaohong Chen & Sokbae Lee & Yuan Liao & Myung Hwan Seo & Youngki Shin & Myunghyun Song, 2023. "SGMM: Stochastic Approximation to Generalized Method of Moments," Papers 2308.13564, arXiv.org, revised Oct 2023.
    6. Lorenzo Rosasco & Silvia Villa & Bang Công Vũ, 2016. "Stochastic Forward–Backward Splitting for Monotone Inclusions," Journal of Optimization Theory and Applications, Springer, vol. 169(2), pages 388-406, May.
    7. Caroline Geiersbach & Teresa Scarinci, 2021. "Stochastic proximal gradient methods for nonconvex problems in Hilbert spaces," Computational Optimization and Applications, Springer, vol. 78(3), pages 705-740, April.
    8. Kengy Barty & Jean-Sébastien Roy & Cyrille Strugarek, 2007. "Hilbert-Valued Perturbed Subgradient Algorithms," Mathematics of Operations Research, INFORMS, vol. 32(3), pages 551-562, August.

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