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