Does the Wiener–Khinchin-Einstein theorem hold for non-Gaussian processes? If so are there any assumptions?

Question

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.

Answer 1

first sentence of Wikipedia article:

In applied mathematics, the Wiener–Khinchin theorem, also known as the Wiener–Khintchine theorem and sometimes as the Wiener–Khinchin–Einstein theorem or the Khinchin–Kolmogorov theorem, states that the autocorrelation function of a wide-sense-stationary random process has a spectral decomposition given by the power spectrum of that process.

(emphasis by me)

So this has nothing to do with Gaussian or not, this has to do with stationarity of your process – a process that is Gaussian for every single point in time isn't necessarily wide-sense stationary.

For example, if you have a random signal with an amplitude that is normal distributed for every instant, but whose variance decreases exponentially with time, you can't apply the Wiener-Khinchin theorem.

If you, on the other hand, have a process that actually only takes discrete values $-1$ and $1$, but does so with constant probability, then you can apply the Wiener-Khinchin theorem.

So:

Does this relationship hold for non-Gaussian processes?

Yes.

And if so are there any assumptions or restrictions in place regarding its use?

The actual restriction is wide-sense stationarity, as you've noticed:

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?

Um, that's not exactly the definition of WSS that I have in mind:

Weak sense stationary means that the autocovariance only depends on the time difference, not on the absolute times. This also means that the autocovariance of zero shift, ie. $E[x(t)x(t)]-\mu(t)\mu(t)$ is constant, which can, by considering that these are two expressions for energy, only happen if the first moment $\mu$ is constant.