  In this paper we show that the limiting distribution of the real and the
imaginary part of the double Fourier transform of a stationary random field is
almost surely an independent vector with Gaussian marginal distributions, whose
variance is, up to a constant, the field's spectral density. The dependence
structure of the random field is general and we do not impose any restrictions
on the speed of convergence to zero of the covariances, or smoothness of the
spectral density. The only condition required is that the variables are adapted
to a commuting filtration and are regular in some sense. The results go beyond
the Bernoulli fields and apply to both short range and long range dependence.
They can be easily applied to derive the asymptotic behavior of the periodogram
associated to the random field. The method of proof is based on new
probabilistic methods involving martingale approximations and also on borrowed
and new tools from harmonic analysis.
