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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?

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    $\begingroup$ Do we assume that $\mathbb E |X(0)| < \infty$? If yes, then, by Fubini, $\mathbb{E} \int_{-T}^T |X(t)| dt = \int_{-T}^T \mathbb{E} |X(t)| dt = 2 T \mathbb{E} |X(0)|$ is finite, and hence $\int_{-T}^T |X(t)| dt$ is finite a.s. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Apr 14, 2018 at 21:54

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Not every stationary stochastic process has a power spectral density, in general it has a power spectral measure. But in any case $X*1_{[-T,+T]}$ is itself a stationary process, whose realisations (smooth or not) are never integrable on $\mathbb R$ (except if $X\equiv 0$). They have a Fourier transform only in the sense of tempered distributions: its values at a point (frequency) $\omega$ are not defined, and taking the expectation of its square doesn't make sense.

Defining a power spectral measure or density for (some) deterministic signals was the main goal of Norbert Wiener's book The Fourier Integral, 1938. Relating PSD of stochastic processes to that (in Wiener's sense) of its realisations is interesting, but it cannot be done by taking the limit of something that doesn't exist...

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