The winding of stationary Gaussian processes

This paper studies the winding of a continuously differentiable Gaussian stationary process $f:\mathbb{R}\to \mathbb{C}$ in the interval $[0,T]$. We give formulae for the mean and the variance of this random variable. The variance is shown to always grow at least linearly with $T$, and conditions for it to be asymptotically linear or quadratic are given. Moreover, we show that if the covariance function together with its second derivative are in $L^2(\mathbb{R})$, then the winding obeys a central limit theorem. These results correspond to similar results for zeroes of real-valued stationary Gaussian functions by Cuzick, Slud and others.