Weakly stationary stochastic processes valued in a separable Hilbert space: Gramian-Cram\'er representations and applications

TL;DR

This paper develops a comprehensive spectral theory for weakly stationary processes in Hilbert spaces, introducing Gramian-Cramér representations and applying them to functional principal component analysis without extra assumptions.

Contribution

It introduces the Gramian-Cramér representation for Hilbert space-valued processes and extends classical spectral results to this setting.

Findings

Established the Gramian-Cramér representation as an isomorphic mapping.

Derived the Cramér-Karhunen-Loève decomposition for Hilbert space processes.

Provided results on the composition and inversion of lag-invariant linear filters.

Abstract

The spectral theory for weakly stationary processes valued in a separable Hilbert space has known renewed interest in the past decade. Here we follow earlier approaches which fully exploit the normal Hilbert module property of the time domain. The key point is to build the Gramian-Cram\'er representation as an isomorphic mapping from the modular spectral domain to the modular time domain. We also discuss the general Bochner theorem and provide useful results on the composition and inversion of lag-invariant linear filters. Finally, we derive the Cram\'er-Karhunen-Lo\`eve decomposition and harmonic functional principal component analysis, which are established without relying on additional assumptions.