Let an autoregressive sequence of order (2left{Y_{t} ; t=0, pm 1, pm 2, ldots ight}) be given

Let an autoregressive sequence of order \(2\left\{Y_{t} ; t=0, \pm 1, \pm 2, \ldots\right\}\) be given by

\[Y_{t}-0.8 Y_{t-1}-0.09 Y_{t-2}=X_{t} ; t=0, \pm 1, \pm 2, \ldots\]

where \(\left\{X_{t} ; t=0, \pm 1, \pm 2, \ldots\right\}\) is the same purely random sequence as in exercise (6.12).

(1) Check whether the sequence \(\left\{Y_{t} ; t=0, \pm 1, \pm 2, \ldots\right\}\) is weakly stationary. If yes, then determine its covariance function and its correlation function.

(2) Sketch its correlation function and compare its graph with the one obtained in exercise (6.12).

Data from Exercise 6.12

Let \(Y_{t}=0.8 Y_{t-1}+X_{t} ; t=0, \pm 1, \pm 2, \ldots\), where \(\left\{X_{t} ; t=0, \pm 1, \pm 2, \ldots\right\}\) is the purely random sequence with parameters \(E\left(X_{t}\right)=0\) and \(\operatorname{Var}\left(X_{t}\right)=1\).

Determine the covariance function and sketch the correlation function of the autoregressive sequence of order \(1\left\{Y_{t} ; t=0, \pm 1, \pm 2, \ldots\right\}\).