A central limit like theorem for Fourier sums

Dominik Janzing; Naji Shajarisales; Michel Besserve

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arXiv · 1707.06819

A central limit like theorem for Fourier sums

Dominik Janzing·Naji Shajarisales·Michel Besserve

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Abstract

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.

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Dominik Janzing, Naji Shajarisales, Michel Besserve. 2017-07-21. A central limit like theorem for Fourier sums. https://arxiv.org/abs/1707.06819

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