A central limit like theorem for Fourier sums

2017/07/21 by Dominik Janzing, Janzing, Dominik, Naji Shajarisales +3
Computer Science · Mathematics · #60Fxx #Bayesian Methods and Mixture Models #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Statistical Methods and Inference

paper · pdf · doi:10.48550/arxiv.1707.06819

openalex publication_date 2017/07/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the probability distributions of values in the complex plane attained by Fourier sums of the form ∑_j=1ⁿ a_j exp(-2πi j nu) /sqrtn 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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