Reference request: Gaussian almost periodic functions

Question

Let $X(x),x\in R^d$, be a stationary gaussian process for which the covariance function $E(X(0)X(x))=C(x)$ is "almost periodic".

Almost periodic means roughly that $C$ is uniformly approximable by trigonometric polynomials. Interesting examples are trigonometric polynomials which are not periodic such as $$C(x)=\frac 12 (cos(x)+cos(\sqrt{2}x)).$$

In the latter example the properties of $X$ are not too hard to study. I was wondering if there was a "general theory" of such processes (ergodicity, ...).

Answer 1

Estimation for almost periodic processes (2006), provides a general method to determine the lines of support of the spectra, with applications to bias and covariance estimation.