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 | References | Similar Articles | Additional Information
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.
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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
Article copyright:
© Copyright 2018
American Mathematical Society