On the real zeros of random trigonometric polynomials with dependent coefficients (1706.01654v1)

Abstract: We consider random trigonometric polynomials of the form [ f_n(t):=\sum_{1\le k \le n} a_{k} \cos(kt) + b_{k} \sin(kt), ] whose entries $(a_{k}){k\ge 1}$ and $(b{k}){k\ge 1}$ are given by two independent stationary Gaussian processes with the same correlation function $\rho$. Under mild assumptions on the spectral function $\psi\rho$ associated with $\rho$, we prove that the expectation of the number $N_n([0,2\pi])$ of real roots of $f_n$ in the interval $[0,2\pi]$ satisfies [ \lim_{n \to +\infty} \frac{\mathbb E\left [N_n([0,2\pi])\right]}{n} = \frac{2}{\sqrt{3}}. ] The latter result not only covers the well-known situation of independent coefficients but allow us to deal with long range correlations. In particular it englobes the case where the random coefficients are given by a fractional Brownian noise with any Hurst parameter.