L^{2}

, because the norm of a tail is at most

∥ X_{0} ∥_{2}

times the corresponding tail sum of

∣ a_{s} ∣

. Its mean is the original mean times

\sum_{s} a_{s}

. Its covariance is time independent and equals

γ_{k}^{Y} = \sum_{s, u \in Z} a_{s} a_{u} γ_{k - s + u} .

Under the summability assumption in (i),

\sum_{k} ∣ γ_{k}^{Y} ∣ \leq (\sum_{s} ∣ a_{s} ∣)^{2} \sum_{k} ∣ γ_{k} ∣ < \infty.

We may therefore interchange all the sums. Setting

h = k - s + u

f_{Y} (ω) = \frac{1}{π} s, u \sum a_{s} a_{u} e^{i (s - u) ω} h \sum γ_{h} e^{ihω} = (s \sum a_{s} e^{i s ω}) (u \sum a_{u} e^{- i u ω}) f_{X} (ω) .

The second factor is the conjugate of the first because the coefficients are real. Hence

f_{Y} (ω) = ∣ α (ω) ∣^{2} f_{X} (ω) .

This spectral density transformation under a linear filter also holds whenever the original spectral measure has a density, without covariance summability: insert its integral representation into the covariance double sum, justified by absolute summability of the filter coefficients.

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