  We consider the winding number of planar stationary Gaussian processes
defined on the line. Under mild conditions, we obtain the asymptotic variance
and the Central Limit Theorem for the winding number as the time horizon tends
to infinity. In the asymptotic regime, our discrete approach is equivalent to
the continuous one studied previously in the literature and our main result
extends the existing ones. Our model allows for a general dependence of the
coordinates of the process and non-differentiability of one of them.
Furthermore, beyond our general framework, we consider as examples an
approximation to the winding number of a process whose coordinates are both
non-differentiable and the winding number of a process which is not exactly
stationary.
