I am new to the study of time series. Recently I have asked a question about the covariance of real and imaginary part of a real(in time domain) stochastic time series and I have received an answer for it. The problem is that for continuous time series the variance of each point in frequency domain is infinite. I've been told there that its the reason that they use power spectrum. Now the confusion comes from the fact that I don't know whether power spectrum is time averaged or not? More precisely as wiki says: $$S_{xx}(\omega)=\lim\limits_{T\to \infty}\mathbf{E} \left[ | \hat{x}_T(\omega) |^2 \right] = \lim\limits_{T\to \infty}\mathbf{E} \left[ \frac{1}{T} \int\limits_0^T x^*(t) e^{i\omega t}\, dt \int\limits_0^T x(t') e^{-i\omega t'}\, dt' \right] = \lim\limits_{T\to \infty}\frac{1}{T} \int\limits_0^T \int\limits_0^T \mathbf{E}\left[x^*(t) x(t')\right] e^{i\omega (t-t')}\, dt\, dt$$ OR $$S_{xx}(\omega)=\mathbf{E} \left[ | \hat{x}_T(\omega) |^2 \right] = \mathbf{E} \left[ \int\limits_0^\infty x^*(t) e^{i\omega t}\, dt \int\limits_0^\infty x(t') e^{-i\omega t'}\, dt' \right] = \int\limits_0^\infty \int\limits_0^\infty \mathbf{E}\left[x^*(t) x(t')\right] e^{i\omega (t-t')}\, dt\, dt$$
For example in Wiener-Khintchine theorem as far as I can see there is no time averaging: $$r_{xx} (\tau) = \int_{-\infty}^\infty S_{xx}(f) e^{2\pi i\tau f} df$$
And is there any difference when the signal is discrete?