General verification of Cramer's condition (C4) for multivariate covariance kernels

Develop general methods to establish Condition (C4) of Cramer's theorem—that the spectral density matrix of a stationary multivariate stochastic process is nonnegative definite for all frequencies—in order to verify the permissibility of proposed multivariate covariance kernels.

Background

For multivariate (vector) stochastic processes, Cramer's theorem specifies permissibility conditions for matrix-valued covariance kernels, including Condition (C4) requiring the spectral density matrix to be nonnegative definite for all frequencies. While auto-spectral permissibility follows from Bochner’s theorem, verifying (C4) for cross-spectral terms is challenging in practice.

The paper notes that, in the absence of general verification tools for (C4), researchers often resort to more restrictive sufficient conditions such as diagonal dominance, or adopt special structures (e.g., separable models) that are easy to check but can be unsuitable for causality analysis. Establishing general methods for (C4) would enable broader, theoretically grounded model construction and validation.