New asymptotics for the mean number of zeros of random trigonometric polynomials with strongly dependent Gaussian coefficients

Abstract

We consider random trigonometric polynomials of the form \[ f_{n}(t):=\frac{1} {\sqrt{n} } \sum _{k=1}^{n}a_{k} \cos (k t)+b_{k} \sin (k t), \] where $(a_{k})_{k\geq 1}$ and $(b_{k})_{k\geq 1}$ are two independent stationary Gaussian processes with the same correlation function $\rho : k \mapsto \cos (k\alpha )$, with $\alpha \geq 0$. We show that the asymptotics of the expected number of real zeros differ from the universal one $\frac{2} {\sqrt{3} }$, holding in the case of independent or weakly dependent coefficients. More precisely, for all $\varepsilon >0$, for all $\ell \in (\sqrt{2} ,2]$, there exists $\alpha \geq 0$ and $n\geq 1$ large enough such that \[ \left |\frac{\mathbb {E} \left [\mathcal {N}(f_{n},[0,2\pi ])\right ]} {n}-\ell \right |\leq \varepsilon , \] where $\mathcal{N} (f_{n},[0,2\pi ])$ denotes the number of real zeros of the function $f_{n}$ in the interval $[0,2\pi ]$. Therefore, this result provides the first example where the expected number of real zeros does not converge as $n$ goes to infinity by exhibiting a whole range of possible subsequential limits ranging from $\sqrt{2} $ to 2.