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

Xiaohong Chen, Department of Economics, London School of Economics & Political Science
Halbert White, 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.

Recommended Citation

Xiaohong Chen and Halbert White (2002) "Asymptotic Properties of Some Projection-based Robbins-Monro Procedures in a Hilbert Space", Studies in Nonlinear Dynamics & Econometrics: Vol. 6: No. 1, Article 1.
http://www.bepress.com/snde/vol6/iss1/art1

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