 Solution

For a weakly stationary process with mean $m$, the autocovariance function is ${\gamma }_k=\mathbb{E}[(X_{t+k}-m)(X_t-m)]$, independent of $t$. The positive definiteness of these covariances gives a unique finite positive spectral measure of a stationary time series on the frequency circle. Its cumulative function is the time-series spectral distribution $F$, with total mass ${\gamma }_0$, andUse one representative of the identified endpoints $-\pi$ and $\pi$ when specifying atoms. For a real-valued process the measure is symmetric, so the imaginary part integrates to zero. If it has a time-series spectral density, thenThe second identity is pointwise for an absolutely summable covariance sequence, as will apply to part (a). Existence of a density alone does not guarantee ordinary pointwise convergence of its Fourier series; in general the reconstruction can be interpreted by its Cesàro means in $L^1$. This fixes the frequency normalization for all three calculations below.