CENTRAL LIMIT AND FUNCTIONAL CENTRAL LIMIT THEOREMS FOR HILBERT-VALUED DEPENDENT HETEROGENEOUS ARRAYS WITH APPLICATIONS

We obtain new CLTs and FCLTs for Hilbert-valued arrays near epoch dependent on mixing processes, as well as new FCLTs for general Hilbert-valued adapted dependent heterogeneous arrays. These theorems are useful in delivering asymptotic distributions for parametric and nonparametric estimators and their functionals in time series econometrics. We give three significant applications for near epoch dependent observations: (1) A new CLT for any plug-in estimator of a cumulative distribution function (e.g., an empirical cdf, or a cdf estimator based on a kernel density estimator), which can in turn deliver distribution results for many Von Mises functionals; (2) A new limiting distribution result for degenerate U-statistics, which delivers distribution results for Bierens' integrated conditional moment tests; (3) A new functional central limit result for Hilbert-valued stochastic approximation procedures, which delivers distribution results for nonparametric recursive generalized method of moment estimators, including nonparametric adaptive learning models.

[1]  Jeffrey M. Wooldridge,et al.  Some Invariance Principles and Central Limit Theorems for Dependent Heterogeneous Processes , 1988, Econometric Theory.

[2]  Donald W. K. Andrews,et al.  A nearly independent, but non-strong mixing, triangular array , 1985, Journal of Applied Probability.

[3]  D. Andrews.  Laws of Large Numbers for Dependent Non-Identically Distributed Random Variables , 1988, Econometric Theory.

[4]  D. Aldous.  The Central Limit Theorem for Real and Banach Valued Random Variables , 1981 .

[5]  G. Pflug,et al.  Stochastic approximation and optimization of random systems , 1992 .

[6]  D. McLeish.  A Maximal Inequality and Dependent Strong Laws , 1975 .

[7]  Yanqin Fan,et al.  Consistent hypothesis testing in semiparametric and nonparametric models for econometric time series , 1999 .

[8]  H. Bierens.  Consistent model specification tests , 1982 .

[9]  Joseph E. Yukich,et al.  Weak Convergence of Smoothed Empirical Processes , 1992 .

[10]  H. White,et al.  A Unified Theory of Estimation and Inference for Nonlinear Dynamic Models , 1988 .

[11]  Xiaohong Chen,et al.  Nonparametric Adaptive Learning with Feedback , 1998 .

[12]  L. Fernholz,et al.  Almost sure convergence of smoothed empirical distribution functions , 1991 .

[13]  N. N. Vakhanii︠a︡.  Probability distributions on linear spaces , 1981 .

[14]  J. Kuelbs.  Probability on Banach spaces , 1978 .

[15]  Charles F. Manski.  Analog Estimation Methods in Econometrics: Chapman & Hall/CRC Monographs on Statistics & Applied Probability , 1988 .

[16]  E. Berger.  Asymptotic behaviour of a class of stochastic approximation procedures , 1986 .

[17]  G. K. Eagleson.  ORTHOGONAL EXPANSIONS AND U‐STATISTICS , 1979 .

[18]  D. McLeish.  On the Invariance Principle for Nonstationary Mixingales , 1977 .

[19]  K. Parthasarathy.  PROBABILITY MEASURES IN A METRIC SPACE , 1967 .

[20]  Charles F. Manski,et al.  Analog estimation methods in econometrics , 1988 .

[21]  Xiaohong Chen,et al.  Laws of Large Numbers for Hilbert Space-Valued Mixingales with Applications , 1996, Econometric Theory.

[22]  J. H. Venter.  On Dvoretzky Stochastic Approximation Theorems , 1966 .

[23]  P. Gaenssler,et al.  On Martingale Central Limit Theory , 1986 .

[24]  Herman J. Bierens,et al.  Asymptotic Theory of Integrated Conditional Moment Tests , 1997 .

[25]  K. Parthasarathy,et al.  Probability measures on metric spaces , 1967 .

[26]  E. Carlstein.  Degenerate U-Statistics Based on Non-Independent Observations , 1988 .

[27]  Ing Rj Ser.  Approximation Theorems of Mathematical Statistics , 1980 .

[28]  Harro Walk.  An invariance principle for the Robbins-Monro process in a Hilbert space , 1977 .

[29]  Adam Jakubowski.  On Limit Theorems for Sums of Dependent Hilbert Space Valued Random Variables , 1980 .

[30]  Central Limit Theorems For Dependent, Heterogeneous Processes With Trending Moments , 1989 .

[31]  Herold Dehling,et al.  Limit theorems for sums of weakly dependent Banach space valued random variables , 1983 .