On the real zeros of random trigonometric polynomials with dependent coefficients

Source abstract

We consider random trigonometric polynomials of the form $$f n ( t ) := ∑ 1 ≤ k ≤ n a k cos ⁡ ( k t ) + b k sin ⁡ ( k t ) , f_n(t):=\sum _{1\le k \le n} a_{k} \cos (kt) + b_{k} \sin (kt),$$ whose coefficients ( a k ) k ≥ 1 (a_{k})_{k\ge 1} and ( b k ) k ≥ 1 (b_{k})_{k\ge 1} are given by two independent stationary Gaussian processes with the same correlation function ρ \rho . Under mild assumptions on the spectral function ψ ρ \psi _\rho associated with ρ \rho , we prove that the expectation of the number N n ( [ 0 , 2 π ] ) N_n([0,2\pi ]) of real roots of f n f_n in the interval [ 0 , 2 π ] [0,2\pi ] satisfies $$lim n → + ∞ E [ N n ( [ 0 , 2 π ] ) ] n = 2 3 . \lim _{n \to +\infty } \frac {\mathbb E\left [N_n([0,2\pi ])\right ]}{n} = \frac {2}{\sqrt {3}}.$$ 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.