Central limit theorem for Fourier transform and periodogram of random fields

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.