Composition of absolutely summable time-series filters ($c_r={\sum }_j{b}_j{a}_{r-j}$)

Two absolutely summable linear filters of a stationary time series compose by convolution: $c_r={\sum }_j{b}_j{a}_{r-j}$ and ${\sum }_r|c_r|\le ({\sum }_r|a_r|)({\sum }_j|b_j|)$. 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.