Define the Fourier transform of an ergodic time series {$x_{t}$} in the wide sense by
$$ X(\omega) = \lim_{T\to\infty}\frac{1}{\sqrt{2T}}\sum_{t=-T}^T x_{t}e^{-i\omega t}.$$
The following relation holds for the power spectrum
$$E[|X(\omega)X(\omega')|]= \begin{cases} S(\omega), & \mbox{if } \omega = \omega' \\ 0, & \mbox{if } \omega \neq \omega' \end{cases}.$$
We need to prove that the probability density of $X(\omega)$ is given by
$$p(X;\theta)=exp{\{-\frac{1}{2}\int\theta(\omega)|X(\omega)|^2d\omega-\psi(\theta)}\}$$
where
$$\theta(\omega) = \frac{1}{S(\omega)}, \\ \psi(\theta) = \frac{1}{2}\int \log(S(\omega))d\omega - \frac{\pi}{2}.$$
And also prove that the KL-divergence between two systems written using their power spectra is
$$KL[S_{1}:S_{2}]=\frac{1}{2\pi}\int_{-\pi}^{\pi}(\frac{S_{1}}{S_{2}} -1 -\log(\frac{S_{1}}{S_{2}}))d\omega.$$
Any suggestion is welcome, thank you.