  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(-2\pi 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.
