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 $p_n(t,Y)=sum_{k=1}^n Y_{k}^1 cos(kt)+Y_{k}^2sin(kt)$ for a given sequence of i.i.d. random variables %${Y_{k,1},Y_{k,2}}_{kge 1}$ $Y^i_{k}$, $iin{1,2}$, $kge 1$, 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 egin{equation*} lim_{n oinfty}rac{ ext{Var}left(mathcal{N}_n([0,pi],Y) ight)}{n}% =lim_{n oinfty}rac{ ext{Var}left(mathcal{N}_n([0,pi],G) ight)}{n}% +rac{1}{30}left(mathbb{E}((Y_{1}^1)^4)-3 ight). end{equation*} 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.