Non universality for the variance of the number of real roots of random trigonometric polynomials

In this article, we consider the following family of random trigonometric polynomials pn(t,Y)=Sigma k=1nYk1cos(kt)+Yk2sin(kt) for a given sequence of i.i.d. random variables Yki, i is an element of{1,2}, k >= 1, which are centered and standardized. We set N([0,pi],Y) the number of real roots over [0,pi] and N([0,pi],G) the corresponding quantity when the coefficients follow a standard Gaussian distribution. We prove under a Doeblin's condition on the distribution of the coefficients that lim (n ->infinity) Var (N-n([0, pi], Y))/n = lim (n ->infinity) Var (N-n([0, pi], G))/n + 1/30 (E ((Y-1(1))(4)) - 3). The latter establishes that the behavior of the variance is not universal and depends on the distribution of the underlying coefficients through their kurtosis. Actually, a more general result is proven in this article, which does not require that the coefficients are identically distributed. The proof mixes a recent result regarding Edgeworth expansions for distribution norms established in Bally et al. (Electron J Probab 23(45):1-51, 2018) with the celebrated Kac-Rice formula.