Central limit theorem for Fourier transform and periodogram of random fields

TL;DR

This paper establishes a central limit theorem for the Fourier transform and periodogram of stationary random fields, demonstrating Gaussian limiting distributions under broad dependence conditions without smoothness or covariance decay restrictions.

Contribution

It introduces a general CLT for Fourier transforms of random fields with minimal assumptions, extending beyond Bernoulli fields and applicable to various dependence structures.

Findings

Limiting distribution of Fourier transform components is Gaussian with spectral density variance.

Results apply to both short-range and long-range dependent fields.

Method combines martingale approximations with harmonic analysis tools.

Abstract

In this paper we show that the limiting distribution of the real and the imaginary part of the double Fourier transform of a stationary random field is almost surely an independent vector with Gaussian marginal distributions, whose variance is, up to a constant, the field's spectral density. The dependence structure of the random field is general and we do not impose any restrictions on the speed of convergence to zero of the covariances, or smoothness of the spectral density. The only condition required is that the variables are adapted to a commuting filtration and are regular in some sense. The results go beyond the Bernoulli fields and apply to both short range and long range dependence. They can be easily applied to derive the asymptotic behavior of the periodogram associated to the random field. The method of proof is based on new probabilistic methods involving martingale approximations and also on borrowed and new tools from harmonic analysis.