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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Real zeros of random trigonometric polynomials with dependent coefficients
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by Jürgen Angst, Thibault Pautrel and Guillaume Poly PDF
Trans. Amer. Math. Soc. 375 (2022), 7209-7260 Request permission

Abstract:

We further investigate the relations between the large degree asymptotics of the number of real zeros of random trigonometric polynomials with dependent coefficients and the underlying correlation function. We consider trigonometric polynomials of the form \[ f_n(t)≔\frac {1}{\sqrt {n}}\sum _{k=1}^{n}a_k \cos (kt)+b_k\sin (kt), ~x\in [0,2\pi ], \] where the sequences $(a_k)_{k\geq 1}$ and $(b_k)_{k\geq 1}$ are two independent copies of a stationary Gaussian process centered with variance one and correlation function $\rho$ with associated spectral measure $\mu _{\rho }$. We focus here on the case where $\mu _{\rho }$ is not purely singular and we denote by $\psi _{\rho }$ its density component with respect to the Lebesgue measure $\lambda$. Quite surprisingly, we show that the asymptotics of the number of real zeros $\mathcal {N}(f_n,[0,2\pi ])$ of $f_n$ in $[0,2\pi ]$ is not related to the decay of the correlation function $\rho$ but instead to the Lebesgue measure of the vanishing locus of $\psi _{\rho }$. Namely, assuming that $\psi _{\rho }$ is $\mathcal {C}^1$ with Hölder derivative on an open set of full measure, one establishes that \[ \lim _{n \to +\infty } \frac {\mathbb {E}\left [\mathcal {N}(f_n,[0,2\pi ])\right ]}{n}= \frac {\lambda (\{\psi _{\rho }=0\})}{\pi \sqrt {2}} + \frac {2\pi - \lambda (\{\psi _{\rho }=0\})}{\pi \sqrt {3}}. \] On the other hand, assuming a sole log-integrability condition on $\psi _{\rho }$, which implies that it is positive almost everywhere, we recover the asymptotics of the independent case: \[ \lim _{n \to +\infty } \frac {\mathbb {E}\left [\mathcal {N}(f_n,[0,2\pi ])\right ]}{n}= \frac {2}{\sqrt {3}}. \] The latter asymptotics thus broadly generalizes the main result of Angst, Dalmao, and Poly [Proc. Amer. Math. Soc. 147 (2019), pp. 205–214] where the spectral density was assumed to be continuous and bounded from below. Besides, with further assumptions of regularity and existence of negative moment for $\psi _{\rho }$, which encompass e.g. the case of random coefficients being increments of fractional Brownian motion with any Hurst parameter, we moreover show that the above convergence in expectation can be strengthened to an almost sure convergence.
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Additional Information
  • Jürgen Angst
  • Affiliation: Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France
  • Email: jurgen.angst@univ-rennes1.fr
  • Thibault Pautrel
  • Affiliation: Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France
  • MR Author ID: 1386743
  • Email: thibault.pautrel@univ-rennes1.fr
  • Guillaume Poly
  • Affiliation: Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France
  • MR Author ID: 997488
  • Email: guillaume.poly@univ-rennes1.fr
  • Received by editor(s): March 14, 2021
  • Received by editor(s) in revised form: February 28, 2022
  • Published electronically: August 10, 2022
  • Additional Notes: This work was supported by the ANR grant UNIRANDOM, ANR-17-CE40-0008.
  • © Copyright 2022 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 375 (2022), 7209-7260
  • MSC (2020): Primary 26C10; Secondary 30C15, 42A05, 60F17, 60G55
  • DOI: https://doi.org/10.1090/tran/8742