Variance of the number of zeros of dependent Gaussian trigonometric polynomials

TL;DR

This paper analyzes the variance of the number of real zeros in dependent Gaussian trigonometric polynomials, showing that dependence does not alter the asymptotic behavior compared to independent cases, and extends results to various models.

Contribution

It provides the first variance asymptotics for dependent Gaussian trigonometric polynomials, generalizing known results from independent cases and utilizing intrinsic properties of the Kac--Rice density.

Findings

Variance asymptotics match the independent case under mild conditions

Extends variance analysis to various models of Gaussian polynomials

Uses Kac--Rice formula without explicit second moment formulas

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 polynomials. Though we use the Kac--Rice formula, we do not use the explicit closed formula for the second moment of the number of zeros, but we rather rely on intrinsic properties of the Kac--Rice density.