  We discuss the probability distributions of Fourier modes of cosmological
density fields using the central limit theorem as it applies to weighted
integrals of random fields. It is shown that if the cosmological principle
holds in a certain sense, i.e., the universe approaches homogeneity and
isotropy sufficiently rapidly on very large scales, the one-point distribution
of each Fourier mode of the density field is Gaussian whether or not the
density field itself is Gaussian. Therefore, one-point distributions of the
power spectrum obtained from observational data or from simulations are not a
good test of whether the density field is Gaussian.
