A central limit like theorem for Fourier sums
Probability
2017-07-24 v1
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.
Cite
@article{arxiv.1707.06819,
title = {A central limit like theorem for Fourier sums},
author = {Dominik Janzing and Naji Shajarisales and Michel Besserve},
journal= {arXiv preprint arXiv:1707.06819},
year = {2017}
}
Comments
7 pages