On the mean square convergence of the convolution representation of linear filters

Mean square convergence is the most frequently considered mode of convergence for the infinite series convolution expressions representing filter outputs in stationary time series analysis. There is confusion, however, also in the literature, about which conditions guarantee that this convergence holds. If only general properties of the input series to the filter are known, it is appropriate to consider the class of series with these properties. For each of several classes of full rank, wide sense stationary, zero-mean, vector time series, a weakest possible condition on the frequency response function of a linear filter is given which guarantees that the time-domain convolution representation of the filter converges to the filter output in mean square, whenever the input series belongs to the class under consideration. The classes considered are (i) the purely nondeterministic series with essentially bounded spectral density matrix, (ii) all purely nondeterministic series, (iii) all series. We then show that more unified resttlts can be obtained if Cesiro sums are utilized to define the convergence of the convolution representation. The mean square convergence of infinite autoregressions is also discussed.