  We consider asymptotic behavior of Fourier transforms of stationary ergodic
sequences with finite second moments. We establish a central limit theorem
(CLT) for almost all frequencies and also an annealed CLT. The theorems hold
for all regular sequences. Our results shed new light on the foundation of
spectral analysis and on the asymptotic distribution of periodogram, and it
provides a nice blend of harmonic analysis, theory of stationary processes and
theory of martingales.
