I would like a validation/critique of the Wiener-Khinchin Theorem which is presented below? Note that I'm a physicists so I may lack rigor. I'm most curious if the general idea is correct, however I am open to hearing about how it could be made more rigorous.
We define the Fourier transform of $g(t)$, $\tilde{g}(f)$ as
\begin{align} \tilde{g}(f) = \int_{t=-\infty}^{+\infty} g(t) e^{-i2\pi f t} dt = \int g(t) e^{-i2\pi f t} dt \end{align}
Integrals without limits specified will be assumed to be from $-\infty$ to $+\infty$.
We have a signal $X(t)$ (random process) which may not be absolutely integrable (for example $X(t) = e^{-i2\pi f_0 t}$).
We introduce a window function $W_1(t)$ which is absolutely integrable and satisfies
$$ \int |W_1(t)|^2 dt = \int |\tilde{W}_1(f)|^2 df = 1 $$
We extend $W_1(t)$ to a scaled family of window function by introducing
$$ W_{\Delta t}(t) = \frac{1}{\Delta t}W_1\left(\frac{t}{\Delta t}\right) $$
$W_{\Delta t}(t)$ are also absolutely integrable and the integral of the squared magnitude is also unity.
The simplest example would be to take
\begin{align} W_1(t) =& \theta(t) \theta(1 - t)\\ W_{\Delta t}(t) =& \frac{1}{\sqrt{\Delta t}} \theta(t) \theta(\Delta t - t) \end{align}
a box window ranging from $0 \le t \le \Delta t$ scaled by $\frac{1}{\sqrt{\Delta t}}$. We define the windowed version of $X(t)$:
\begin{align} X_{\Delta t}(t) = X(t)W_{\Delta t}(t) \end{align}
The Power Spectral Density of $X(t)$ is given by
\begin{align} S_{XX}(f) = \lim_{\Delta t \rightarrow \infty} \langle |\tilde{X}_{\Delta t}(f)|^2 \rangle \end{align}
Where $\langle \cdot \rangle$ indicates taking the expectation value of the random process.
The auto-correlation function of $X(t)$ is given by
\begin{align} R_{XX}(t_1, t_2) = \langle X(t_1)X^*(t_2) \rangle \end{align}
We consider stationary processes which satisfy
\begin{align} R_{XX}(t_1, t_2) = R_{XX}(t_1 - t_2, 0) = R_{XX}(t_1 - t_2) \end{align}
The Wiener-Khinchin theorem states that for a stationary process $X(t)$ we have that the power spectral density is the Fourier transform of the auto-correlation function
\begin{align} S_{XX}(f) = \tilde{R}_{XX}(f) = \int R_{XX}(\tau) e^{-i2\pi f \tau} d\tau \end{align}
This is proven as follows:
\begin{align} S_{XX}(f) =& \lim_{\Delta t \rightarrow \infty} \langle |\tilde{X}_{\Delta t}(f)|^2 \rangle\\ =&\lim_{\Delta t \rightarrow \infty} \int \int \langle X(t_1)X^*(t_2) \rangle W_{\Delta t}(t_1)W_{\Delta t}(t_2) e^{-i2\pi f t_1} e^{+i2\pi f t_2} dt_1 dt_2\\ =&\lim_{\Delta t \rightarrow \infty} \int \int R_{XX}(t_1-t_2) W_{\Delta t}(t_1)W_{\Delta t}(t_2) e^{-i2\pi f t_1} e^{+i2\pi f t_2} dt_1 dt_2\\ \end{align}
we can now do a change of variables using $t_1 = \tau + t_2$ on the inner integral
\begin{align} =& \lim_{\Delta t \rightarrow \infty} \int \int R_{XX}(\tau) e^{-i 2\pi f \tau} W_{\Delta t}(\tau + t_2)W_{\Delta t}(t_2) d\tau dt_2\\ \end{align}
We now interchange the integrals
\begin{align} =& \lim_{\Delta t \rightarrow \infty} \int \left(R_{XX}(\tau) e^{-i 2\pi f \tau}\right) \left(\int W_{\Delta t}(\tau + t_2)W_{\Delta t}(t_2) dt_2\right) d\tau\\ \end{align}
The proof would be complete if we could pass the limit in and show
\begin{align} \lim_{\Delta t \rightarrow \infty}\left(\int W_{\Delta t}(\tau + t_2)W_{\Delta t}(t_2) dt_2\right) = 1 \end{align}
I think this is done without too much trouble.. We can pass the limit into this second integral and apply the definition of $W_{\Delta t}(t)$:
\begin{align} \int \lim_{\Delta t \rightarrow \infty} \frac{1}{\Delta t} W_1\left(\frac{\tau}{\Delta t} + \frac{t_2}{\Delta t}\right)W_1^*\left(\frac{t_2}{\Delta t}\right) dt_2 \end{align}
But as $\Delta t \rightarrow \infty$ we have $\frac{\tau}{\Delta t}\rightarrow 0$ so we get
\begin{align} =& \int \lim_{\Delta t \rightarrow \infty} \frac{1}{\Delta t} W_1\left(\frac{t_2}{\Delta t}\right)W_1^*\left(\frac{t_2}{\Delta t}\right) dt_2\\ =& \lim_{\Delta t \rightarrow \infty}\int W_{\Delta t}(t_2)W^*_{\Delta t}(t_2) dt_2\\ =& 1 \end{align}
By the normalization of $W_{\Delta t}(t)$ specified above.
This proof seems to check out from my physics perspective. How does it look to others? Is the general idea roughly valid, in particular the bits at the end involving actually taking the $\Delta t$ limit? Perhaps it is obvious to someone else exactly what constraints are necessary on $X(t)$ and $R_{XX}(t)$ or $W_1(t)$ for this all to work out?