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On the real zeros of random trigonometric polynomials with dependent coefficients. (English) Zbl 1406.26007
Authors’ abstract: We consider random trigonometric polynomials of the form $f_n(t):=\sum _{1\leq k \leq n} a_k \cos (kt) + b_k \sin (kt),$ whose coefficients $$(a_k)_{k\geq 1}$$ and $$(b_k)_{k\geq 1}$$ 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 $\lim\limits_{n \rightarrow +\infty } \frac {\mathbb{E}\left [N_n([0,2\pi ])\right ]}{n} = \frac {2}{\sqrt {3}}.$
The latter result not only covers the well-known situation of independent coefficients but it also allows us to deal with long-range correlations. In particular, it includes the case where the random coefficients are given by a fractional Brownian noise with any Hurst parameter.

##### MSC:
 26C10 Real polynomials: location of zeros 42A05 Trigonometric polynomials, inequalities, extremal problems 12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems) 30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
longmemo
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##### References:
 [1] Jean-Marc Aza\“\i s, Federico Dalmao, Jos\'”e Le\'on, Ivan Nourdin, and Guillaume Poly, Local universality of the number of zeros of random trigonometric polynomials with continuous coefficients, arXiv:1512.05583, 2015. [2] Aza\“\i s, Jean-Marc; Dalmao, Federico; Le\'”on, Jos\'e R., CLT for the zeros of classical random trigonometric polynomials, Ann. Inst. Henri Poincar\'e Probab. Stat., 52, 2, 804-820 (2016) · Zbl 1342.60025 [3] Aza\“\i s, Jean-Marc; Le\'”on, Jos\'e R., CLT for crossings of random trigonometric polynomials, Electron. J. Probab., 18, no. 68, 17 pp. (2013) · Zbl 1284.60048 [4] J\“urgen Angst and Guillaume Poly, Universality of the mean number of real zeros of random trigonometric polynomials under a weak cram\'”er condition, arXiv:1511.08750, 2015. [5] Bary, N. K., A treatise on trigonometric series. Vols. I, II, Authorized translation by Margaret F. Mullins. A Pergamon Press Book, Vol. I: xxiii+553 pp. Vol. II: xix+508 pp. (1964), The Macmillan Co., New York [6] Beran, Jan, Statistics for long-memory processes, Monographs on Statistics and Applied Probability 61, x+315 pp. (1994), Chapman and Hall, New York · Zbl 0869.60045 [7] Borichev, Alexander; Nishry, Alon; Sodin, Mikhail, Entire functions of exponential type represented by pseudo-random and random Taylor series, J. Anal. Math., 133, 361-396 (2017) · Zbl 1387.30031 [8] Dunnage, J. E. A., The number of real zeros of a random trigonometric polynomial, Proc. London Math. Soc. (3), 16, 53-84 (1966) · Zbl 0141.15003 [9] Edelman, Alan; Kostlan, Eric, How many zeros of a random polynomial are real?, Bull. Amer. Math. Soc. (N.S.), 32, 1, 1-37 (1995) · Zbl 0820.34038 [10] Erd\"os, Paul; Offord, A. C., On the number of real roots of a random algebraic equation, Proc. London Math. Soc. (3), 6, 139-160 (1956) · Zbl 0070.01702 [11] Farahmand, Kambiz, On the average number of real roots of a random algebraic equation, Ann. Probab., 14, 2, 702-709 (1986) · Zbl 0609.60074 [12] Farahmand, K., On the variance of the number of real zeros of a random trigonometric polynomial, J. Appl. Math. Stochastic Anal., 10, 1, 57-66 (1997) · Zbl 0880.60058 [13] Farahmand, Kambiz, Topics in random polynomials, Pitman Research Notes in Mathematics Series 393, x+163 pp. (1998), Longman, Harlow · Zbl 0949.60010 [14] Flasche, Hendrik, Expected number of real roots of random trigonometric polynomials, Stochastic Process. Appl., 127, 12, 3928-3942 (2017) · Zbl 1377.60063 [15] Glendinning, Richard, The growth of the expected number of real zeros of a random polynomial, J. Austral. Math. Soc. Ser. A, 46, 1, 100-121 (1989) · Zbl 0671.60063 [16] Grafakos, Loukas, Classical Fourier analysis, Graduate Texts in Mathematics 249, xvi+489 pp. (2008), Springer, New York · Zbl 1220.42001 [17] Granville, Andrew; Wigman, Igor, The distribution of the zeros of random trigonometric polynomials, Amer. J. Math., 133, 2, 295-357 (2011) · Zbl 1218.60042 [18] Iksanov, Alexander; Kabluchko, Zakhar; Marynych, Alexander, Local universality for real roots of random trigonometric polynomials, Electron. J. Probab., 21, Paper No. 63, 19 pp. (2016) · Zbl 1361.30009 [19] Ibragimov, I. A.; Maslova, N. B., The average number of zeros of random polynomials, Vestnik Leningrad. Univ., 23, 19, 171-172 (1968) · Zbl 0235.60060 [20] Ibragimov, I. A.; Maslova, N. B., The mean number of real zeros of random polynomials. I. Coefficients with zero mean, Teor. Verojatnost. i Primenen., 16, 229-248 (1971) · Zbl 0277.60051 [21] Kac, M., On the average number of real roots of a random algebraic equation, Bull. Amer. Math. Soc., 49, 314-320 (1943) · Zbl 0060.28602 [22] Kac, M., On the average number of real roots of a random algebraic equation. II, Proc. London Math. Soc. (2), 50, 390-408 (1949) · Zbl 0033.14702 [23] Littlewood, J. E.; Offord, A. C., On the Number of Real Roots of a Random Algebraic Equation, J. London Math. Soc., 13, 4, 288-295 (1938) · Zbl 0020.13604 [24] Renganathan, N.; Sambandham, M., On the average number of real zeros of a random trigonometric polynomial with dependent coefficients. II, Indian J. Pure Appl. Math., 15, 9, 951-956 (1984) · Zbl 0553.60063 [25] Sambandham, M., On the number of real zeros of a random trigonometric polynomial, Trans. Amer. Math. Soc., 238, 57-70 (1978) · Zbl 0379.60060 [26] Zygmund, A., Trigonometric series: Vols. I, II, Second edition, reprinted with corrections and some additions, Vol. I. xiv+383 pp.; Vol. II: vii+364 pp. (two volumes bound as one) pp. (1968), Cambridge University Press, London-New York
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