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 $p_n(t,Y)=\sum_{k=1}^n Y_{k,1} \cos(kt)+Y_{k,2}\sin(kt)$ for a given sequence of i.i.d. random variables $\{Y_{k,1},Y_{k,2}\}_{k\ge 1}$ which are centered and standardized. We set $\mathcal{N}([0,\pi],Y)$ the number of real roots over $[0,\pi]$ and $\mathcal{N}([0,\pi],G)$ 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 $$ \lim_{n\to\infty}\frac{\text{Var}\left(\mathcal{N}_n([0,\pi],Y)\right)}{n} =\lim_{n\to\infty}\frac{\text{Var}\left(\mathcal{N}_n([0,\pi],G)\right)}{n} +\frac{1}{30}\left(\mathbb{E}(Y_{1,1}^4)-3\right). $$ 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's expansions for distribution norms established in arXiv:1606.01629 with the celebrated Kac-Rice formula.