Weakly stationary stochastic processes valued in a separable Hilbert space: Gramian-Cramér representations and applications

The spectral theory for weakly stationary processes valued in a separable Hilbert space has known renewed interest in the past decade. However, the recent literature on this topic is often based on restrictive assumptions or lacks important insights. In this paper, we follow earlier approaches which fully exploit the normal Hilbert module property of the space of Hilbert-valued random variables. This approach clarifies and completes the isomorphic relationship between the modular spectral domain to the modular time domain provided by the Gramian-Cram\'er representation. 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 without relying on simplifying assumptions.