We consider (complex) Gaussian analytic functions on a horizontal strip, whose distribution is invariant with respect to horizontal shifts (i.e., "stationary"). Let N(T) be the number of zeroes in [0,T] x [a,b]. We present an extension of a result by Wiener, concerning the existence and characterization of the limit N(T)/T as T approaches infinity, as well as characterize the growth of the variance of N(T). We then discuss analogues of these results in a few other settings, such as zeroes of real-analytic Gaussian functions and winding of planar Gaussian functions, pointing out interesting similarities and differences. For the last part, we consider the "persistence probability" of real Gaussian functions (i.e., the probability that a function has no zeroes at all in some interval). We give simple conditions for this probability to be exponential, and explaining why these lead to "i.i.d.-like" behavior. Based in part on joint works with Jeremiah Buckley and Ohad Feldheim.