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 $(a_k{)}_{k\ge 1}$ and $(b_k{)}_{k\ge 1}$ are two independent stationary Gaussian processes with the same correlation function $\rho :k\mapsto \cos (k\alpha )$, with $\alpha \ge 0$. 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 $\epsilon >0$, for all $\ell \in (\sqrt{2},2]$, there exists $\alpha \ge 0$ and $n\ge 1$ large enough such that $$\left|\frac{\mathbb{E}\left[\mathcal{N}(f_n,[0,2\pi ])\right]}{n}-\ell \right|\le \epsilon ,$$ 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 do not converge as $n$ goes to infinity by exhibiting a whole range of possible limits ranging from $\sqrt{2}$ to 2.

16 pages, 2 figures