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Proceedings of the American Mathematical Society

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On the real zeros of random trigonometric polynomials with dependent coefficients


Authors: Jürgen Angst, Federico Dalmao and Guillaume Poly
Journal: Proc. Amer. Math. Soc. 147 (2019), 205-214
MSC (2010): Primary 26C10; Secondary 30C15, 42A05
DOI: https://doi.org/10.1090/proc/14216
Published electronically: October 3, 2018
MathSciNet review: 3876743
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Abstract: We consider random trigonometric polynomials of the form

$\displaystyle f_n(t):=\sum _{1\le k \le n} a_{k} \cos (kt) + b_{k} \sin (kt), $

whose coefficients $ (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

$\displaystyle \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 allows us to deal with longrange correlations. In particular, it includes the case where the random coefficients are given by a fractional Brownian noise with any Hurst parameter.

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Additional Information

Jürgen Angst
Affiliation: Institut de recherche mathématiques de Rennes, Université de Rennes 1 35042, Rennes 1, France
Email: jurgen.angst@univ-rennes1.fr

Federico Dalmao
Affiliation: Departamento de Matematico y Estadistica del Litoral, Universidad de la Republica, 25 de agosto 281, 50000 Salto, Uruguay
Email: fdalmao@unorte.edu.uy

Guillaume Poly
Affiliation: Institut de recherche mathématiques de Rennes, Université de Rennes 1 35042, Rennes 1, France
Email: guillaume.poly@univ-rennes1.fr

DOI: https://doi.org/10.1090/proc/14216
Received by editor(s): June 6, 2017
Received by editor(s) in revised form: April 3, 2018
Published electronically: October 3, 2018
Additional Notes: This work was supported by the ANR grant UNIRANDOM
Communicated by: David Levin
Article copyright: © Copyright 2018 American Mathematical Society