The Wiener–Khinchin-Einstein theorem states that the auto-correlation $(r_{xx}(\tau))$ and spectral density $(S(f))$ are Fourier duals, i.e.
$$r_{xx}(\tau) = \int^{+\infty}_{-\infty} S(f) \exp\left( 2\pi i \tau f \right)\:df$$
This relationship has several assumptions:
-The process must be stationary, i.e the spatial correlation depend only upon the the distance between two points ($\tau$) and not the orientation ($\vec{\tau}$). For Gaussian processes this requires the first and second order moments to be stationary.
My question is:
- Does this relationship hold for non-Gaussian processes?
- And if so are there any assumptions or restrictions in place regarding its use? For if it holds for non-Gaussian processes, is it suffient for the process Weak or wide-sense stationarity, i.e. only the first and second order moment need to be stationary and higher order moments can be non-stationary, or are there more strict conditions? My gut feeling is the former since the equation requires no-higher order information, however I'm not sure if that is the case.