Asymptotic Properties of Some Projection-Based Robbins-Monro Procedures in a Hilbert Space
U of California, Economics Working Paper No. 2002-07
77 Pages Posted: 5 Jun 2002
Date Written: January 2002
Let H be an infinite-dimensional real separable Hilbert space. Given an unknown mapping M : H 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 mix-ingale, -dependent error processes.
Keywords: Recursive Estimation, Non-Parametric Estimation, Hilbert Space, Generalized Method of Moments
JEL Classification: C14
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