Identifying the Spectral Representation of Hilbertian Time Series

TL;DR

This paper establishes consistent estimation of the spectral representation of covariance operators in Hilbertian time series, extending previous methods and providing a rigorous foundation for PCA in infinite-dimensional spaces.

Contribution

It generalizes existing spectral estimation methods for Hilbertian time series and offers a simple proof of a key property of centered random elements in Hilbert spaces.

Findings

Square-root n consistency in spectral estimation

Extension of Bathia et al. (2010) method

Rigorous formulation of PCA in Hilbert spaces

Abstract

We provide square-root n consistency results regarding estimation of the spectral representation of covariance operators of Hilbertian time series, in a setting with imperfect measurements. This is a generalization of the method developed in Bathia et al. (2010). The generalization relies on an important property of centered random elements in a separable Hilbert space, namely, that they lie almost surely in the closed linear span of the associated covariance operator. We provide a straightforward proof to this fact. This result is, to our knowledge, overlooked in the literature. It incidentally gives a rigorous formulation of PCA in Hilbert spaces.