Linear filter of a stationary time series ($Y_t={\sum }_s{a}_s{X}_{t-s}$)

An absolutely summable sequence of coefficients defines $Y_t={\sum }_s{a}_s{X}_{t-s}$ in mean square. It remains weakly stationary and has covariance ${\gamma }_k^Y={\sum }_{s,u}{a}_s{a}_u{\gamma }_{k-s+u}$. Its frequency response is $\alpha (\omega )={\sum }_s{a}_s{e}^{is\omega }$.

Filtering multiplies the time-series spectral density by the squared modulus of the frequency response. Insert the spectral integral into the covariance double sum and interchange sums using absolute summability of the filter. This holds even when the input covariances are not absolutely summable, provided the spectral measure has a density.