Quenched limit theorems for Fourier transforms and periodogram

TL;DR

This paper establishes quenched central limit theorems for the Fourier transform and periodogram of stationary ergodic and Markov processes, without requiring irreducibility or regularity conditions, using harmonic analysis and ergodic theory.

Contribution

It proves quenched CLTs for Fourier transforms of stationary processes under minimal assumptions, extending understanding of spectral analysis in non-irreducible Markov processes.

Findings

Quenched CLT holds for Fourier transforms of stationary ergodic processes.

Results apply to Markov chains starting from arbitrary points.

Conditions for quenched CLT without centering are characterized.

Abstract

In this paper, we study the quenched central limit theorem for the discrete Fourier transform. We show that the Fourier transform of a stationary ergodic process, suitable centered and normalized, satisfies the quenched CLT conditioned by the past sigma algebra. For functions of Markov chains with stationary transitions, this means that the CLT holds with respect to the law of the chain started at a point for almost all starting points. It is necessary to emphasize that no assumption of irreducibility with respect to a measure or other regularity conditions are imposed for this result. We also discuss necessary and sufficient conditions for the validity of quenched CLT without centering. The results are highly relevant for the study of the periodogram of a Markov process with stationary transitions which does not start from equilibrium. The proofs are based of a nice blend of harmonic analysis, theory of stationary processes, martingale approximation and ergodic theory.