Spectral representation theorem for a stationary time series ($X_t={\int }_{-1/2}^{1/2}{e}^{2\pi it\omega }dZ(\omega )$)

A centered weakly stationary process has a representation $X_t={\int }_{-1/2}^{1/2}{e}^{2\pi it\omega }dZ(\omega )$ with orthogonal random increments. Their variance measure is the spectral measure. When it has density $f$, the autocovariance is $\gamma (h)=\int e^{2\pi ih\omega }f(\omega )d\omega$.