Spectral representation theorem for a stationary time series
= Spectral representation theorem for a stationary time series
{title2=$X_t=\int_{-1/2}^{1/2}e^{2\pi it\omega}\,dZ(\omega)$}
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 $\gamma(h)=\int e^{2\pi ih\omega}f(\omega)\,d\omega$.