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

Xiaohong Chen 1 , 1  and Halbert White 2 , 2
  • 1 Department of Economics, London School of Economics & Political Science
  • 2 Department of Economics, University of California, San Diego, hwhite@albert.ucsd.edu

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.

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SNDE recognizes that advances in statistics and dynamical systems theory can increase our understanding of economic and financial markets. The journal seeks both theoretical and applied papers that characterize and motivate nonlinear phenomena. Researchers are required to assist replication of empirical results by providing copies of data and programs online. Algorithms and rapid communications are also published.

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