The winding of stationary Gaussian processes

TL;DR

This paper analyzes the winding number of stationary Gaussian processes, providing formulas for its mean and variance, conditions for asymptotic growth, and a central limit theorem under specific covariance conditions.

Contribution

It introduces new formulas and conditions for the mean, variance, and distributional limits of the winding number of Gaussian processes.

Findings

Variance grows at least linearly with time

Conditions for asymptotic linear or quadratic variance growth

Winding obeys a central limit theorem under certain covariance conditions

Abstract

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.