Variance of the number of zeroes of shift-invariant Gaussian analytic functions

Following Wiener, we consider the zeroes of Gaussian analytic functions in a strip in the complex plane, with translation-invariant distribution. We show that the variance of the number of zeroes in a long horizontal rectangle [−T,T] × [a, b] is asymptotically between cT and CT2, with positive constants c and C. We also supply with conditions (in terms of the spectral measure) under which the variance grows asymptotically linearly with T, as a quadratic function of T, or has intermediate growth. The results are compared with known results for real stationary Gaussian processes and other models.