Linear filter of a stationary time series
= Linear filter of a stationary time series
{title2=$Y_t=\sum_s a_sX_{t-s}$}
An absolutely summable sequence of coefficients defines $Y_t=\sum_s a_sX_{t-s}$ in mean square. It remains weakly stationary and has <covariance> $\gamma_k^Y=\sum_{s,u}a_sa_u\gamma_{k-s+u}$. Its frequency response is $\alpha(\omega)=\sum_sa_se^{is\omega}$.