Given a stationary integrable stocastic process $x(t)$ we can build the spectrum $$S_T(\omega)=\frac{1}{2\pi T}\left| \int_0^T\mathrm{d}t\, x(t)\,\mathrm{e}^{\mathrm{i}\omega t}\right|^2,$$ which can be considered as a $T,\omega$ indexed stochastic variable.
The Wiener-Khinchin theorem, under some assumptions, says something like (in one of its forms) $$S_T(\omega)\rightarrow\mathcal{FFT}\Big\{\mathbb{E}[x(t)\,x(0)]\Big\}\quad\text{for}\quad T\rightarrow\infty\tag1$$
where $\mathbb{E}[x(t)x(0)]$ is the autocorrelation function and $\mathcal{FFT}$ indicates a Fourier Transform.
I would like to know better the precise kind of convergence in (1):
- Is it maybe simply intended that $\mathbb E[S_T(\omega)] \rightarrow\mathcal{FFT}\Big\{\mathbb E[x(t)\,x(0)]\Big\}$ for $T\rightarrow \infty$, or is some other convergence involved?
- If (1) is the case, what do we know about the distribution of $S_T(\omega)$? For example, does the variance of $S_T(\omega)$ go to zero for large $T$?
I would like to have primarily a picture in the continuous case. References to books where to find the answer are also welcome, as I have never been able to understand this point.