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2020 New asymptotics for the mean number of zeros of random trigonometric polynomials with strongly dependent Gaussian coefficients
Pautrel Thibault
Electron. Commun. Probab. 25: 1-13 (2020). DOI: 10.1214/20-ECP314

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.

Citation

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Pautrel Thibault. "New asymptotics for the mean number of zeros of random trigonometric polynomials with strongly dependent Gaussian coefficients." Electron. Commun. Probab. 25 1 - 13, 2020. https://doi.org/10.1214/20-ECP314

Information

Received: 25 February 2020; Accepted: 20 April 2020; Published: 2020
First available in Project Euclid: 8 May 2020

zbMATH: 1445.60052
MathSciNet: MR4095048
Digital Object Identifier: 10.1214/20-ECP314

Subjects:
Primary: 26C10
Secondary: 30C15, 42A05

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