## Electronic Communications in Probability

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

Pautrel Thibault

#### 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.

#### Article information

Source
Electron. Commun. Probab., Volume 25 (2020), paper no. 36, 13 pp.

Dates
Accepted: 20 April 2020
First available in Project Euclid: 8 May 2020

https://projecteuclid.org/euclid.ecp/1588903422

Digital Object Identifier
doi:10.1214/20-ECP314

#### Citation

Thibault, Pautrel. New asymptotics for the mean number of zeros of random trigonometric polynomials with strongly dependent Gaussian coefficients. Electron. Commun. Probab. 25 (2020), paper no. 36, 13 pp. doi:10.1214/20-ECP314. https://projecteuclid.org/euclid.ecp/1588903422

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