On the real zeros of random trigonometric polynomials with dependent coefficients
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- by Jürgen Angst, Federico Dalmao and Guillaume Poly PDF
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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 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 \[ \lim _{n \to +\infty } \frac {\mathbb E\left [N_n([0,2\pi ])\right ]}{n} = \frac {2}{\sqrt {3}}. \] 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.References
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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
- MR Author ID: 946948
- Email: fdalmao@unorte.edu.uy
- Guillaume Poly
- Affiliation: Institut de recherche mathématiques de Rennes, Université de Rennes 1 35042, Rennes 1, France
- MR Author ID: 997488
- Email: guillaume.poly@univ-rennes1.fr
- 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
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 205-214
- MSC (2010): Primary 26C10; Secondary 30C15, 42A05
- DOI: https://doi.org/10.1090/proc/14216
- MathSciNet review: 3876743