Proof. (i) Since \(\gamma (-\tau ) = \gamma (\tau )\), the terms for \(\tau \) and \(-\tau \) pair up as \(\gamma (\tau )(e^{-i\omega \tau } + e^{i\omega \tau }) = 2\gamma (\tau )\cos (\omega \tau )\), which is real and even in \(\omega \).

(ii) For integers \(m\), \(\int _{-\pi }^{\pi }e^{i\omega m}\,d\omega \) is \(2\pi \) if \(m = 0\) and \(0\) otherwise, since \(e^{i\omega m}\) then completes \(|m|\) whole turns. So \begin {align*} \int _{-\pi }^{\pi }f(\omega )e^{i\omega \tau }\,d\omega &= \frac {1}{2\pi }\sum _k\gamma (k)\int _{-\pi }^{\pi }e^{i\omega (\tau -k)}\,d\omega \\ &= \frac {1}{2\pi }\gamma (\tau )\cdot 2\pi = \gamma (\tau ) , \end {align*}

where exchanging sum and integral is justified by the absolute convergence of \(\sum |\gamma (k)|\). Put \(\tau = 0\) for the variance. □

Proof. As for an MA process, \(\gamma (\tau ) = \sigma _Z^2\sum _j\psi _j\psi _{j+\tau }\) (with \(\psi _j = 0\) for \(j < 0\), this holds for all \(\tau \)). Then \begin {align*} \sum _\tau \gamma (\tau )e^{-i\omega \tau } &= \sigma _Z^2\sum _j\sum _\tau \psi _j\psi _{j+\tau }e^{-i\omega (j+\tau )}e^{i\omega j} \\ &= \sigma _Z^2\Bigl (\sum _k\psi _ke^{-i\omega k}\Bigr )\Bigl (\sum _j\psi _je^{i\omega j}\Bigr ) , \end {align*}

with \(k = j + \tau \). The second factor is the complex conjugate of the first, so the product is \(|\psi (e^{-i\omega })|^2\). For an ARMA process \(\psi (B) = \theta (B)/\phi (B)\). □

Solution.

White noise

Here \(\gamma (0) = \sigma ^2\) and \(\gamma (\tau ) = 0\) otherwise, so only one term survives: \[ f(\omega ) = \frac {\sigma ^2}{2\pi } , \] constant in \(\omega \). Every frequency contributes equally, which is what the name means, by analogy with white light.

AR(1)

By Theorem 16.1.3 with \(\phi (z) = 1 - \alpha z\), and \(|1 - \alpha e^{-i\omega }|^2 = (1 - \alpha e^{-i\omega })(1 - \alpha e^{i\omega }) = 1 - 2\alpha \cos \omega + \alpha ^2\), \[ f(\omega ) = \frac {\sigma ^2}{2\pi }\cdot \frac {1}{|1 - \alpha e^{-i\omega }|^2} = \frac {\sigma ^2}{2\pi (1 - 2\alpha \cos \omega + \alpha ^2)} . \] (Summing the two geometric series in the definition, with \(\gamma (\tau ) = \sigma ^2\alpha ^{|\tau |}/(1 - \alpha ^2)\), gives the same.) For \(\alpha > 0\) this is largest at \(\omega = 0\): the series is dominated by slow movement, which is what positive autocorrelation means. For \(\alpha < 0\) the peak moves to \(\omega = \pi \), the fastest representable oscillation, alternation from one observation to the next.

MA(1)

With \(\theta (z) = 1 + \beta z\), \(|1 + \beta e^{-i\omega }|^2 = 1 + 2\beta \cos \omega + \beta ^2\), so \[ f(\omega ) = \frac {\sigma ^2}{2\pi }(1 + 2\beta \cos \omega + \beta ^2) , \] a single smooth hump, at \(\omega = 0\) if \(\beta > 0\) and at \(\omega = \pi \) if \(\beta < 0\). Note that \(\beta \) and \(1/\beta \) give spectra of the same shape, as they give the same ac.f. (Section 7).

Solution. Here \(\phi (z) = 1 - z + \frac 12z^2\), and \begin {align*} |\phi (e^{-i\omega })|^2 &= \bigl (1 - e^{-i\omega } + \tfrac 12e^{-2i\omega }\bigr )\bigl (1 - e^{i\omega } + \tfrac 12e^{2i\omega }\bigr ) \\ &= 1 + 1 + \tfrac 14 - 2\cos \omega + \cos 2\omega - \cos \omega \\ &= \tfrac 94 - 3\cos \omega + \cos 2\omega . \end {align*}

The spectrum is largest where this is smallest. With \(c = \cos \omega \) and \(\cos 2\omega = 2c^2 - 1\), it is \(2c^2 - 3c + \frac 54\), minimised at \(c = \frac 34\). So the peak is at \(\omega = \arccos 0.75 = 0.723\) radians, a period of \(2\pi /0.723 = 8.7\) observations, where \(|\phi |^2 = 0.125\) and \(f = 8\sigma ^2/(2\pi )\), against \(4\sigma ^2/(2\pi )\) at \(\omega = 0\) and \(0.16\sigma ^2/(2\pi )\) at \(\omega = \pi \). The ac.f. oscillates with period exactly \(8\); the two periods are close but not equal, because the damping of the ac.f. shifts the peak of the spectrum (Figure 14, panel 4).