On the real zeros of random trigonometric polynomials with dependent coefficients

TL;DR

This paper analyzes the asymptotic behavior of the expected number of real zeros of random trigonometric polynomials with dependent Gaussian coefficients, extending known results to include long-range correlations such as fractional Brownian noise.

Contribution

It establishes the limiting ratio of expected zeros for polynomials with dependent Gaussian coefficients, including cases with long-range dependence like fractional Brownian noise.

Findings

Expected zeros ratio converges to 2/√3 as degree increases.

Results include cases with long-range dependence, such as fractional Brownian noise.

Generalizes classical results for independent coefficients to dependent Gaussian processes.

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.