We obtain new central limit theorems (CLT's)
and functional central limit theorems (FCLT's) for
Hilbert-valued arrays near epoch dependent on mixing processes,
and also new FCLT's 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
(c.d.f.) (e.g., an empirical c.d.f., or a c.d.f. 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's
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.