Quenched Asymptotics for the Discrete Fourier Transforms of a Stationary Process

TL;DR

This paper extends the Central Limit Theorem and Invariance Principle for Discrete Fourier Transforms of stationary processes to the quenched setting, highlighting the necessity of random normalization and exploring convergence properties.

Contribution

It introduces the quenched versions of classical results for Fourier transforms, discusses the importance of random normalization, and provides new insights into convergence in distribution in metric spaces.

Findings

Quenched CLT and Invariance Principle hold under certain conditions.

Random normalization is necessary for these quenched results.

Reduced test functions simplify the analysis of quenched convergence.

Abstract

In this dissertation, we show that the Central Limit Theorem and the Invariance Principle for Discrete Fourier Transforms discovered by Peligrad and Wu can be extended to the quenched setting. We show that the random normalization introduced to extend these results is necessary and we discuss its meaning. We also show the validity of the quenched Invariance Principle for fixed frequencies under some conditions of weak dependence. In particular, we show that this result holds in the martingale case. The discussion needed for the proofs allows us to show some general facts apparently not noticed before in the theory of convergence in distribution. In particular, we show that in the case of separable metric spaces the set of test functions in the Portmanteau theorem can be reduced to a countable one, which implies that the notion of quenched convergence, given in terms of convergence a.s. of conditional expectations, specializes in the right way in the regular case when the state space is metrizable and second-countable. We also collect and organize several disperse facts from the existing theory in a consistent manner towards the statistical spectral analysis of the Discrete Fourier Transforms, providing a comprehensive introduction to topics in this theory that apparently have not been systematically addressed in a self-contained way by previous references.