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We consider the winding number of planar stationary Gaussian processesdefined on the line. Under mild conditions, we obtain the asymptotic varianceand the Central Limit Theorem for the winding number as the time horizon tendsto infinity. In the asymptotic regime, our discrete approach is equivalent tothe continuous one studied previously in the literature and our main resultextends the existing ones. Our model allows for a general dependence of thecoordinates of the process and non-differentiability of one of them.Furthermore, beyond our general framework, we consider as examples anapproximation to the winding number of a process whose coordinates are bothnon-differentiable and the winding number of a process which is not exactlystationary.

Studying the winding number for stationary Gaussian processes using real variables