On the CLT for discrete Fourier transforms of functional time series

TL;DR

This paper investigates the weak convergence of discrete Fourier transforms for stationary, ergodic functional time series in Hilbert spaces, establishing conditions for CLT and weak convergence.

Contribution

It provides new sharp conditions for the weak convergence of Fourier transforms and a CLT for partial sums in the context of functional time series.

Findings

Established weak convergence conditions for Fourier transforms

Proved a CLT for partial sums under mild assumptions

Enhanced understanding of spectral analysis in functional data

Abstract

We consider a strictly stationary and ergodic sequence of random elements taking values in some Hilbert space. Our target is to study the weak convergence of the discrete Fourier transforms under sharp conditions. As a side-result we obtain the regular CLT for partial sums under mild assumptions.