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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)
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