I tried asking this question on stackexchange and I also extensively researched it online without results so I will ask here.

In my textbook the Wiener-Khinchin theorem is used to connect the auto-correlation definition of PSD with an "intuitive interpretation" of power spectral density for deterministic signals. It says:

[S_xx(\omega) = \lim_{T\rightarrow\infity \frac{1}{2T}E[|FT{X(t)*I_{[-T,T]}}|^2]]

But such a theorem assumes you can take the fourier transform of a(truncated) realization of the random process, which unless I am missing something - may not be the case. Such a realization may not be integrable.

So what is going on? Is there a different definition of integral which allows you to do this?

I tried asking this question on stackexchange and I also extensively researched it online without results so I will ask here.

In my textbook the Wiener-Khinchin theorem is used to connect the auto-correlation definition of PSD with an "intuitive interpretation" of power spectral density for deterministic signals. It says:

$S_{xx}(\omega) = \lim_{T\rightarrow\infty} \frac{1}{2T}\mathbb E\left[|FT\{X(t)*I_{[-T,T]}\}|^2\right]$

But such a theorem assumes you can take the fourier transform of a(truncated) realization of the random process, which unless I am missing something - may not be the case. Such a realization may not be integrable.

So what is going on? Is there a different definition of integral which allows you to do this?