# Stochastic process with Bessel function autocorrelation. (Rayleigh (Jakes) fading for radiowave propagation)

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 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.

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.