Have the following stochastic process $f(t)$ been studied in mathematics ?
It is stationary, Gaussian, $f(t)-$complex independent Gaussians $N(0,1)$.
The autocorrelation is given by the
 zero-order Bessel function of the first kind: $J\_{0} (\tau)$.


In radio wave propagation it is called [Rayleigh fading](http://en.wikipedia.org/wiki/Jakes_fading_model) or sometimes Jakes fading model.
And it is often used in signal processing.
So I wonder that it might be some studies of this process in mathematics,
which might give me some new point of view on it.


In particular I hope for the following: There should be some natural and mathematically clearly formulated reason (model) which will lead to Bessel function auto-correlation.
In signal processing this is known as "radio wave amplitudes" autocorrelate with Bessel function. But can we avoid "radio waves" ? Can we just formulate some simple mathematical model from which we can derive this autocorrelation from something like a central limit theorem or some other clear mathematical reason. 
I think this should be known, but I am not expert in the field.



Stochastic process with Bessel function autocorrelation. (Rayleigh (Jakes) fading for radiowave propagation)
Jakes is a bit beyond me, but I'm fairly sure that the Bessel function $J\_0$ emerges, because of the integral presentation
$$
J\_0(x)=\frac1{2\pi}\int\_{-\pi}^{\pi}e^{ix\sin t}dt.
$$
If you have two plane waves of the same frequency, the other reflected so that the two copies have angular separation $\theta$. Then their correlation would vary like $e^{ix\cos\theta}$, because the the projection of the wavelength of the other wave along the direction of propagation of the other gets multiplied by $\cos\theta$. Now treat $\theta$ as a random variable uniformly distributed over the circle and compute the average.


IOW, I think that the Bessel function emerges as a consequence of the underlying geometry as opposed to being a design parameter. Hopefully a more knowledgeable person can answer. This was just a bit too long to fit into a comment.


