A central limit like theorem for Fourier sums

TL;DR

This paper proves a central limit theorem for Fourier sums with random frequencies, showing their distributions approach a 2D Gaussian under certain conditions on the coefficients.

Contribution

It establishes a new CLT-like result for Fourier sums with random frequencies, extending classical probabilistic limit theorems to this setting.

Findings

Fourier sums' distributions converge to a 2D Gaussian

Convergence holds for coefficients with finite third moment

Results apply under various pseudometrics on distributions

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.