= Spectral measure of a random harmonic oscillation
{title2=$\mathcal F=\tfrac12\delta_{-\omega_0}+\tfrac12\delta_{\omega_0}$}
For uncorrelated zero-mean unit-variance amplitudes $A,B$, the process $A\cos(\omega_0t)+B\sin(\omega_0t)$ has covariance $\cos(\omega_0k)$ 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.
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