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 $$Y^i_{k}$$ , $$i\in \{1,2\}$$ , $$k\ge 1$$ , 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 $$\begin{aligned} \lim _{n\rightarrow \infty }\frac{\text {Var}\left( {\mathcal {N}}_n([0,\pi ],Y)\right) }{n} =\lim _{n\rightarrow \infty }\frac{\text {Var}\left( {\mathcal {N}}_n([0,\pi ],G)\right) }{n} +\frac{1}{30}\left( {\mathbb {E}}\left( \left( Y_{1}^1\right) ^4\right) -3\right) . \end{aligned}$$ 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.