I've encountered the following relation a few times (Risken: The Fokker-Planck Equation Methods of Solution and Applications and Gardiner: Handbook of Stochastic Methods):
\begin{equation} \langle \hat{x}(f_1)\hat{x}(f_2)\rangle = \delta(f_1 + f_2) S(f_1) \end{equation}
where $x$ is a stochastic stationary process and $S(f)$ it's power spectrum, defined as:
\begin{equation} S(f) = \lim_{T \rightarrow \infty} \frac{1}{T}\langle |\hat{x_T}(f)|^2 \rangle \end{equation}
where $x_T(t) = x(t)$ for $t \in [-T/2,T/2]$ and $x_T(t) = 0$ otherwise.
In the derivation of this relation, the assumption is made that the Fourier transform and it's inverse exist:
\begin{equation} \hat{x}(f) = \int_{-\infty}^{\infty}dt e^{-2\pi ift}x(t) \end{equation}
which for a stationary signal by definition is not the case and is also the reason why in the derivation of the Wiener-Khinchin theorem the windowed $x_T$ is normally considered first and only in the end the limit $T \rightarrow \infty$ is taken.
My question is two part:
To what degree is this only an issue with regards to precise mathematical rigor? Put antoher way, are there some known cases when applying the above relation could backfire?
Is it possible to avoid issues with the assumption about the existence of the Fourier transform by taking the approach used in deriving the Wiener-Khinchin theorem of using the windowed $x_T$ followed by limit $T \rightarrow \infty $? (outlined below)
Assuming:
\begin{equation} \hat{x}(f) = \lim_{T \rightarrow \infty} \int_{-T/2}^{T/2} dt e^{-2\pi i t f} x(t) \end{equation}
we can write
\begin{equation} \langle \hat{x}(f_1)\hat{x}(f_2)\rangle = \lim_{T_1 \rightarrow \infty} \lim_{T_2 \rightarrow \infty} \int_{-T_1/2}^{T_1/2} \int_{-T_2/2}^{T_2/2} dt_1 dt_2 e^{-2\pi i (t_1 f_1 + t_2 f_2)} \langle x(t_1)x(t_2) \rangle \end{equation}
Introducing variable $\tau = t_2 - t_1$:
\begin{equation} \langle \hat{x}(f_1)\hat{x}(f_2)\rangle = \lim_{T_1 \rightarrow \infty} \lim_{T_2 \rightarrow \infty} \int_{-T_1/2}^{T_1/2} dt_1 e^{-2\pi i t_1 (f_1 + f_2)} \int_{-T_2/2 - t_1}^{T_2/2 - t_1} d\tau e^{-2\pi i f_2 \tau} \langle x(t_1)x(t_1 + \tau) \rangle \end{equation}
Applying the limits separately:
\begin{equation} \langle \hat{x}(f_1)\hat{x}(f_2)\rangle = \lim_{T_1 \rightarrow \infty} \int_{-T_1/2}^{T_1/2} dt_1 e^{-2\pi i t_1 (f_1 + f_2)} \lim_{T_2 \rightarrow \infty}\int_{-T_2/2 - t_1}^{T_2/2 - t_1} d\tau e^{-2\pi i f_2 \tau} \langle x(t_1)x(t_1 + \tau) \rangle \end{equation}
For the second limit at fixed finite $t_1$, with $T \rightarrow \infty$, the integral will approach the Fourier transform of the correlation function, which per Wiener-Khinchin theorem equals to the power spectrum $S(f)$.
\begin{equation} \langle \hat{x}(f_1)\hat{x}(f_2)\rangle = \lim_{T_1 \rightarrow \infty} \int_{-T_1/2}^{T_1/2} dt_1 e^{-2\pi i t_1 (f_1 + f_2)} S(f_2) \end{equation}
Assuming that:
\begin{equation} \delta(f) = \lim_{T \rightarrow \infty} \int_{-T/2}^{T/2} dt e^{2\pi i f t} \end{equation}
it follows:
\begin{equation} \langle \hat{x}(f_1)\hat{x}(f_2)\rangle = \delta(f_1 + f_2) S(f_2) \end{equation}