I am following the arguments presented in the paper AN-255 Power Spectra Estimation, from Texas Instruments, to learn how to derive the power spectral density for a stationary stochastic process, and do not understand one of their steps. I am looking at Page 4 only of the the document for the purposes of this question.
The derivation is done by considering only a single realisation (sample function), denoted by $x(t)$, of the stochastic process. Then they do the usual thing of only considering a truncated version of this signal $x_T(t)$, which allows the Fourier transform of it to exist, denoted by $X_T(f)$ (because one of the requirements for FT to exist is absolute integrability, which isn't satisfied for random signals).
I am confused with how they justify taking the limit $T\rightarrow\infty$ in the right hand column of page 4, shown here:
I understand that you cannot take the limit $T\rightarrow\infty$ directly on the truncated sample function's Fourier transform $X_T(f)$, because it is not defined in that limit (indeed, that was the reason for doing the truncation in the first place).
I also understand that $X_T(f)$ results from a single realisation only of the stochastic process, and if you were to repeat the experiment again you would obtain a different $X_T(f)$. As such, this function is actually a random variable itself (at each frequency $f$), and therefore it makes sense to do an ensemble average over many realisations, and consequently motivates taking an expectation.
What I don't understand is why the expectation solves the problem, and allows you to take the limit $T\rightarrow\infty$ after you have taken the expectation:
What is it about the expectation of $X_T(f)$ that means the limit $T\rightarrow\infty$ can then be taken? I'm having a hard time seeing it... Could someone spell this out plainly for me?