A central limit like theorem for Fourier sums

We consider the probability distributions of values in the complex plane attained by Fourier sums of the form sum_{j=1}^n a_j exp(-2pi i j nu) /sqrt{n} when the frequency nu is drawn uniformly at random from an interval of length 1. If the coefficients a_j are i.i.d. drawn with finite third moment, the distance of these distributions to an isotropic two-dimensional Gaussian on C converges in probability to zero for any pseudometric on the set of distributions for which the distance between empirical distributions and the underlying distribution converges to zero in probability.