A note on Herglotz's theorem for time series on function spaces

TL;DR

This paper extends Herglotz's theorem to Hilbert-valued time series, enabling frequency domain analysis and spectral representation for complex functional processes, including those with jumps and long memory.

Contribution

It generalizes Herglotz's theorem to operator-valued measures for functional time series, facilitating spectral analysis and finite-dimensional reduction under weaker assumptions.

Findings

Proves Herglotz's theorem for Hilbert-valued time series.

Establishes existence of a functional Cramér representation for broad process classes.

Provides an optimal finite-dimensional reduction method.

Abstract

In this article, we prove Herglotz's theorem for Hilbert-valued time series. This requires the notion of an operator-valued measure, which we shall make precise for our setting. Herglotz's theorem for functional time series allows to generalize existing results that are central to frequency domain analysis on the function space. In particular, we use this result to prove the existence of a functional Cram{\'e}r representation of a large class of processes, including those with jumps in the spectral distribution and long-memory processes. We furthermore obtain an optimal finite dimensional reduction of the time series under weaker assumptions than available in the literature. The results of this paper therefore enable Fourier analysis for processes of which the spectral density operator does not necessarily exist.