ON THE REAL ZEROS OF RANDOM TRIGONOMETRIC POLYNOMIALS WITH DEPENDENT COEFFICIENTS

We consider random trigonometric polynomials of the form f(n)(t) := Sigma(1 <= k <= n) a(k) cos(kt) + b(k) sin(kt), whose coefficients (a(k))(k > 1) and (b(k))(k>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 ->+infinity) E[N-n([0,2 pi])]/n = 2/root 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.