  In this paper, we study the quenched central limit theorem for the discrete
Fourier transform. We show that the Fourier transform of a stationary ergodic
process, suitable centered and normalized, satisfies the quenched CLT
conditioned by the past sigma algebra. For functions of Markov chains with
stationary transitions, this means that the CLT holds with respect to the law
of the chain started at a point for almost all starting points. It is necessary
to emphasize that no assumption of irreducibility with respect to a measure or
other regularity conditions are imposed for this result. We also discuss
necessary and sufficient conditions for the validity of quenched CLT without
centering. The results are highly relevant for the study of the periodogram of
a Markov process with stationary transitions which does not start from
equilibrium. The proofs are based of a nice blend of harmonic analysis, theory
of stationary processes, martingale approximation and ergodic theory.
