Generation of the special linear group by elementary matrices in some measure Banach algebras

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.

Automated summary

We study the variance asymptotics of real zeros of trigonometric polynomials with random dependent Gaussian coefficients and find that, under mild conditions, they have the same asymptotic behavior as in the independent framework. Furthermore, our proof goes beyond this framework and explicitly determines the variance asymptotics of various random Gaussian process models. Our approach relies on the intrinsic properties of the Kac-Rice density, which allows for a concise and elegant proof of the variance asymptotics.