Filtering a Stationary Time Series

In the first part of the MAT8181 course, on linear (univariate) time series, I forgot to mention an important theorem. Let  be a stationary time series, and  a sequence of real numbers such that

then the time series  defined as

is a stationary time series. Further, one can get easily that

This result can be used, if necessary in the exercises (that might save some time actually). I did not include this property in my notes because it is a bit technical to establish that this sum exists, and that the time series is stationary. It is rather simple with the spectral density (since  where  stands for the filter generating function), but I did not mention the spectral density since it requires some knowledge on Fourier analysis…