## On the real zeros of random trigonometric polynomials with dependent coefficients

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- by Jürgen Angst, Federico Dalmao and Guillaume Poly PDF
- Proc. Amer. Math. Soc.
**147**(2019), 205-214 Request permission

## 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

- Jean-Marc Azaïs, Federico Dalmao, José León, Ivan Nourdin, and Guillaume Poly,
*Local universality of the number of zeros of random trigonometric polynomials with continuous coefficients*, arXiv:1512.05583, 2015. - Jean-Marc Azaïs, Federico Dalmao, and José R. León,
*CLT for the zeros of classical random trigonometric polynomials*, Ann. Inst. Henri Poincaré Probab. Stat.**52**(2016), no. 2, 804–820 (English, with English and French summaries). MR**3498010**, DOI 10.1214/14-AIHP653 - Jean-Marc Azaïs and José R. León,
*CLT for crossings of random trigonometric polynomials*, Electron. J. Probab.**18**(2013), no. 68, 17. MR**3084654**, DOI 10.1214/EJP.v18-2403 - Jürgen Angst and Guillaume Poly,
*Universality of the mean number of real zeros of random trigonometric polynomials under a weak cramér condition*, arXiv:1511.08750, 2015. - N. K. Bary,
*A treatise on trigonometric series. Vols. I, II*, A Pergamon Press Book, The Macmillan Company, New York, 1964. Authorized translation by Margaret F. Mullins. MR**0171116** - Jan Beran,
*Statistics for long-memory processes*, Monographs on Statistics and Applied Probability, vol. 61, Chapman and Hall, New York, 1994. MR**1304490** - Alexander Borichev, Alon Nishry, and Mikhail Sodin,
*Entire functions of exponential type represented by pseudo-random and random Taylor series*, J. Anal. Math.**133**(2017), 361–396. MR**3736496**, DOI 10.1007/s11854-017-0037-0 - J. E. A. Dunnage,
*The number of real zeros of a random trigonometric polynomial*, Proc. London Math. Soc. (3)**16**(1966), 53–84. MR**192532**, DOI 10.1112/plms/s3-16.1.53 - Alan Edelman and Eric Kostlan,
*How many zeros of a random polynomial are real?*, Bull. Amer. Math. Soc. (N.S.)**32**(1995), no. 1, 1–37. MR**1290398**, DOI 10.1090/S0273-0979-1995-00571-9 - Paul Erdös and A. C. Offord,
*On the number of real roots of a random algebraic equation*, Proc. London Math. Soc. (3)**6**(1956), 139–160. MR**73870**, DOI 10.1112/plms/s3-6.1.139 - Kambiz Farahmand,
*On the average number of real roots of a random algebraic equation*, Ann. Probab.**14**(1986), no. 2, 702–709. MR**832032** - K. Farahmand,
*On the variance of the number of real zeros of a random trigonometric polynomial*, J. Appl. Math. Stochastic Anal.**10**(1997), no. 1, 57–66. MR**1437951**, DOI 10.1155/S1048953397000051 - Kambiz Farahmand,
*Topics in random polynomials*, Pitman Research Notes in Mathematics Series, vol. 393, Longman, Harlow, 1998. MR**1679392** - Hendrik Flasche,
*Expected number of real roots of random trigonometric polynomials*, Stochastic Process. Appl.**127**(2017), no. 12, 3928–3942. MR**3718101**, DOI 10.1016/j.spa.2017.03.018 - Richard Glendinning,
*The growth of the expected number of real zeros of a random polynomial*, J. Austral. Math. Soc. Ser. A**46**(1989), no. 1, 100–121. MR**966287** - Loukas Grafakos,
*Classical Fourier analysis*, 2nd ed., Graduate Texts in Mathematics, vol. 249, Springer, New York, 2008. MR**2445437** - Andrew Granville and Igor Wigman,
*The distribution of the zeros of random trigonometric polynomials*, Amer. J. Math.**133**(2011), no. 2, 295–357. MR**2797349**, DOI 10.1353/ajm.2011.0015 - Alexander Iksanov, Zakhar Kabluchko, and Alexander Marynych,
*Local universality for real roots of random trigonometric polynomials*, Electron. J. Probab.**21**(2016), Paper No. 63, 19. MR**3563891**, DOI 10.1214/16-EJP9 - I. A. Ibragimov and N. B. Maslova,
*The average number of zeros of random polynomials*, Vestnik Leningrad. Univ.**23**(1968), no. 19, 171–172 (Russian, with English summary). MR**0238376** - I. A. Ibragimov and N. B. Maslova,
*The mean number of real zeros of random polynomials. I. Coefficients with zero mean*, Teor. Verojatnost. i Primenen.**16**(1971), 229–248 (Russian, with English summary). MR**0286157** - M. Kac,
*On the average number of real roots of a random algebraic equation*, Bull. Amer. Math. Soc.**49**(1943), 314–320. MR**7812**, DOI 10.1090/S0002-9904-1943-07912-8 - M. Kac,
*On the average number of real roots of a random algebraic equation. II*, Proc. London Math. Soc. (2)**50**(1949), 390–408. MR**30713**, DOI 10.1112/plms/s2-50.5.390 - J. E. Littlewood and A. C. Offord,
*On the Number of Real Roots of a Random Algebraic Equation*, J. London Math. Soc.**13**(1938), no. 4, 288–295. MR**1574980**, DOI 10.1112/jlms/s1-13.4.288 - N. Renganathan and M. Sambandham,
*On the average number of real zeros of a random trigonometric polynomial with dependent coefficients. II*, Indian J. Pure Appl. Math.**15**(1984), no. 9, 951–956. MR**761283** - M. Sambandham,
*On the number of real zeros of a random trigonometric polynomial*, Trans. Amer. Math. Soc.**238**(1978), 57–70. MR**461648**, DOI 10.1090/S0002-9947-1978-0461648-4 - A. Zygmund,
*Trigonometric series: Vols. I, II*, Cambridge University Press, London-New York, 1968. Second edition, reprinted with corrections and some additions. MR**0236587**

## 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