A central limit like theorem for Fourier sums
Abstract
We consider the probability distributions of values in the complex plane attained by Fourier sums of the form Σj=1n aj exp(-2π i j nu) /sqrtn when the frequency nu is drawn uniformly at random from an interval of length 1. If the coefficients aj 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.
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