Asymptotic properties of recursivem-estimators in an infinite-dimensional Hilbert spac

Many economic phenomena, such as adaptive learning and forecasting, display complicated nonlinear dynamics with and without feedback. Parametric recursive m-estimation procedures are subject to misspecification since economic theory rarely yields precise functional forms. Current nonparametric recursive m-estimation procedures are restricted to cases that have linear functional form, almost independent, additive noise and no feedback. My dissertation provides asymptotic theorems to handle stochastic nonlinear dynamics with dependent, non-additive noise in a functional space.;In my first chapter, I introduce the concepts of Hilbert space-valued $Lsb{p}$-mixingales and NED functions, which constitute a broad class of dependent heterogeneous processes commonly seen in economic time series. New weak and strong laws of large numbers are established for such processes. A useful exponential inequality is also obtained for Hilbert space-valued martingale difference processes. In the second chapter, I obtain new central and functional central limit theorems for Hilbert space-valued NED functions of some general functional space-valued mixing processes. I apply these results in my third and fourth chapters to establish asymptotic properties for the nonparametric recursive m-estimators.;My third chapter proposes sieve-type Hilbert space-valued stochastic approximation procedures. In particular, they are procedures with finite-dimensional projections and random truncations. These procedures are attractive because they are easy to compute, allows non-linearity, imposes no prior bound on the value of the estimator, permits non-additive and Hilbert space-valued mixingale error processes. I obtain asymptotic properties including almost-sure convergence in norm, asymptotic normality, law of iterated logarithm and mean rate of convergence. All these properties are established under conditions weaker than those required for the current available results. The law of iterated logarithm is new even when restricted to multivariate stochastic approximation framework. One can apply these results to Monte Carlo simulation, nonparametric recursive m-estimation and forecasting. In the fourth chapter, I modify the third chapter's estimation procedures to incorporate feedback. Almost-sure convergence in the weak topology is obtained under conditions similar to the weakest possible conditions for multivariate SA with feedback. These procedures are suitable for studying economic agents' adaptive learning behaviors.