The winding of stationary Gaussian processes

This paper studies the winding of a continuously differentiable Gaussian stationary process \(f:\mathbb R\rightarrow \mathbb {C}\) 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 Malevich, Cuzick, Slud and others.