Composition of absolutely summable time-series filters
= Composition of absolutely summable time-series filters
{title2=$c_r=\sum_j b_ja_{r-j}$}
Two absolutely summable <linear filters of a stationary time series> compose by <convolution>: $c_r=\sum_jb_ja_{r-j}$ and $\sum_r|c_r|\leq(\sum_r|a_r|)(\sum_j|b_j|)$. Absolute convergence in the space of square-integrable <random variables> justifies exchanging the sums. The frequency responses multiply, giving the <spectral density transformation under a linear filter> twice.