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 p n ( t , Y ) = ∑ k = 1 n Y k 1 cos ( k t ) + Y k 2 sin ( k t ) for a given sequence of i.i.d. random variables Y k i , i ∈ { 1 , 2 } , k ≥ 1 , which are centered and standardized. We set N ( [ 0 , π ] , Y ) the number of real roots over [ 0 , π ] and N ( [ 0 , π ] , 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 → ∞ Var N n ( [ 0 , π ] , Y ) n = lim n → ∞ Var N n ( [ 0 , π ] , 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.