Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 47 / 2 / c

The zero means of both amplitudes give $\mathbb{E}{X}_t=0$. Their unit variances and zero covariance give, for arbitrary integer $s,t$,Thus the variance is one and the covariance depends only on lag, proving weak stationarity. Independence or normality of the amplitudes is unnecessary. Sincethe spectral measure of a random harmonic oscillation assigns mass $1/2$ to each of $-{\omega }_0,{\omega }_0$. The corresponding right-continuous time-series spectral distribution isThese spectral atoms represent a persistent oscillation. There is no ordinary absolutely continuous spectral density for this process.