  In this dissertation, we show that the Central Limit Theorem and the
Invariance Principle for Discrete Fourier Transforms discovered by Peligrad and
Wu can be extended to the quenched setting. We show that the random
normalization introduced to extend these results is necessary and we discuss
its meaning. We also show the validity of the quenched Invariance Principle for
fixed frequencies under some conditions of weak dependence. In particular, we
show that this result holds in the martingale case.
  The discussion needed for the proofs allows us to show some general facts
apparently not noticed before in the theory of convergence in distribution. In
particular, we show that in the case of separable metric spaces the set of test
functions in the Portmanteau theorem can be reduced to a countable one, which
implies that the notion of quenched convergence, given in terms of convergence
a.s. of conditional expectations, specializes in the right way in the regular
case when the state space is metrizable and second-countable.
  We also collect and organize several disperse facts from the existing theory
in a consistent manner towards the statistical spectral analysis of the
Discrete Fourier Transforms, providing a comprehensive introduction to topics
in this theory that apparently have not been systematically addressed in a
self-contained way by previous references.
