Variance of the number of zeros of dependent Gaussian trigonometric polynomials

Abstract:

We compute the variance asymptotics for the number of real zeros of trigonometric polynomials with random dependent Gaussian coefficients and show that under mild conditions, the asymptotic behavior is the same as in the independent framework. In fact our proof goes beyond this framework and makes explicit the variance asymptotics of various models of random Gaussian processes. Our proof relies on intrinsic properties of the Kac–Rice density in order to give a short and concise proof of variance asymptotics.