Central limit theorem for Fourier transforms of stationary processes

TL;DR

This paper proves a central limit theorem for Fourier transforms of stationary ergodic sequences, enhancing understanding of spectral analysis and periodogram distribution in time series analysis.

Contribution

It establishes a CLT for Fourier transforms at almost all frequencies for regular sequences, linking harmonic analysis with martingale theory.

Findings

CLT holds for Fourier transforms of stationary ergodic sequences.

Results apply to all regular sequences.

Provides new insights into spectral analysis and periodogram asymptotics.

Abstract

We consider asymptotic behavior of Fourier transforms of stationary ergodic sequences with finite second moments. We establish a central limit theorem (CLT) for almost all frequencies and also an annealed CLT. The theorems hold for all regular sequences. Our results shed new light on the foundation of spectral analysis and on the asymptotic distribution of periodogram, and it provides a nice blend of harmonic analysis, theory of stationary processes and theory of martingales.