Spectral measure of a random harmonic oscillation ($\mathcal{F}=\frac{1}{2}{\delta }_{-{\omega }_0}+\frac{1}{2}{\delta }_{{\omega }_0}$)

For uncorrelated zero-mean unit-variance amplitudes $A,B$, the process $A\cos ({\omega }_{0}t)+B\sin ({\omega }_{0}t)$ has covariance $\cos ({\omega }_{0}k)$ and hence the displayed two-atom spectral measure of a stationary time series. Independence or Gaussian amplitudes are unnecessary for weak stationarity. A phase shift need not preserve the whole joint amplitude distribution, so strict stationarity cannot be inferred from these second moments alone.