(a) Show that the ACF of a moving average of a white noise process \(x_{t}=\frac{1}{k} \sum_{j=1}^{k} w_{t-j}\) is 0 outside the window of the moving average; that is, for \(x_{s}\) and \(x_{t}\) in a \(k\)-moving average of a white noise, \(ho(s, t)=0\) if \(|s-t| \geq k\).

(b) Show that the ACF of a moving average of a white noise process with \(k=3\) is as given in equation (5.52).

\(ho(s, t)= \begin{cases}1 & \text { for } s=t, \\ \frac{2}{3} & \text { for }|s-t|=1, \\ \frac{1}{3} & \text { for }|s-t|=2, \\ 0 & \text { for }|s-t| \geq 3 .\end{cases} \tag{5.52}\)