Electronic Communications in Probability

New asymptotics for the mean number of zeros of random trigonometric polynomials with strongly dependent Gaussian coefficients

Pautrel Thibault

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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.

Article information

Source
Electron. Commun. Probab., Volume 25 (2020), paper no. 36, 13 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1588903422

Digital Object Identifier
doi:10.1214/20-ECP314

Subjects
Primary: 26C10: Polynomials: location of zeros [See also 12D10, 30C15, 65H05]
Secondary: 30C15: Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) {For algebraic theory, see 12D10; for real methods, see 26C10} 42A05: Trigonometric polynomials, inequalities, extremal problems

Keywords
random trigonometric polynomials nodal set strong dependence

Rights
Creative Commons Attribution 4.0 International License.

Citation

Thibault, Pautrel. New asymptotics for the mean number of zeros of random trigonometric polynomials with strongly dependent Gaussian coefficients. Electron. Commun. Probab. 25 (2020), paper no. 36, 13 pp. doi:10.1214/20-ECP314. https://projecteuclid.org/euclid.ecp/1588903422


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References

  • [ADL] J.M Azaïs, F. Dalmao, J.R Leon, I. Nourdin and G. Poly, Local universality of the number of zeros of random trigonometric polynomials with continuous coefficients, arXiv:1512.05583.
  • [ADP19] J. Angst, F. Dalmao and G. Poly, On the real zeros of random trigonometric polynomials with dependent coefficients, Proc. Amer. Math. Soc. 147 (2019), no. 1, 205–214.
  • [AP15] J. Angst and G. Poly, Universality of the mean number of real zeros of random trigonometric polynomials under a weak cramer condition, arXiv:1511.08750, 2015
  • [AW09] J-M. Azaïs and M. Wschebor, Level sets and extrema of random processes and fields, Chap. 3 p71.
  • [DNV18] Y. Do, O. Nguyen and V. Vu, Roots of random polynomials with coefficients of polynomial growth, Ann. Probab., Vol 46, Number 5, 2407-2494, 2018.
  • [Dun66] J.E.A. Dunnage, The number of real zeros of a random trigonometric polynomial, Proc. London Math. Soc. (3), 16:53-84, 1966
  • [Far86] K. Farahmand, On the average number of real roots of a random algebraic equation, Ann. Prob., 14(2):702-709, 1986.
  • [FL12] K. Farahmand and T. Li, Real zeros of three different cases of polynomials with random coefficients, Rocky Mountain J. Math. 42 (2012), 1875-1892.
  • [Fla17] H. Flasche, Expected number of real roots of random trigonometric polynomials, Stochastic Processes and their Applications, pages -, 2017
  • [IKM16] A. Iksanov, Z. Kabluchko and A. Marynych, Local universality for real roots of random trigonometric polynomials, Electron. J. Probab., Vol 21, paper no 63,19 pp, 2016
  • [IM68] I.A. Ibragimov and N.B. Maslova, On the expected number of real zeros of random polynomials i. coefficients with zero means, Theory of Probability and Its Applications, 16(2):228-248, 1971.
  • [Mat10] J. Matayoshi, The real zeros of a random algebraic polynomial with dependent coefficients, Rocky Mountain J. Math. 42, No 3, pp. 1015-1034
  • [Muk18] S. Mukeru, Average number of real zeros of random algebraic polynomials defined by the increments of fractional Brownian motion, J. Th. Prob., p.1-23, 2018
  • [NNV15] H. Nguyen, O. Nguyen and V.Vu, On the number of real roots of random polynomials, Com. in Cont. Math. Vol 18, 1550052, 2015.
  • [Kac43] M. Kac. On the average number of real roots of a random algebraic equation, Bull. Amer. Math. Soc. 49:314-320, 1943
  • [Pir19a] A. Pirhadi, Real zeros of random trigonometric polynomials with pairwise equal blocks of coefficients, arXiv:1905.13349, 2019
  • [Pir19b] A. Pirhadi, Real zeros of random cosine polynomials with palindromic blocks of coefficients, arXiv:1908.08154,to appear in Rocky Mountain J. Math., 2020
  • [RS84] N. Renganathan and M. Sambandham, On the average number of real zeros of a random trigonometric polynomial with dependent coefficients, Indian Journal of Pure and Applied Mathematrics, 15(9):951-956, 1984
  • [Sam78] M. Sambandham, On the number of real zeros of a random trigonometric polynomial, Trans. Amer. Math. Soc., 238:57-70, 1978