AR data PDF, Large $N$ (circular process)

Evaluation of (10.7) is very fast itself. If this is not fast enough, it can be greatly simplified in the large-$N$ case. In particular, we can approximate the PDF evaluated using the AR parameter estimates

$$\log p({\bf x}; \hat{\sigma}^2, \hat{\bf a}) = -N/2 \log(2\pi\hat{\sigma}^2) - N/2;$$

and the log-determinent of the FIM as

$$\log {\rm det} ({\bf I}(\sigma^2,a_1,a_2 \ldots a_P)) = P \log N + \log\left({N \over 2 \sigma^4} \right)$$

Problem 3   Show the above.