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

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.