Covariance of the number of real zeros of a random trigonometric polynomial

For random coefficients a j and b j we consider a random trigonometric polynomial defined as T n ( θ ) = ∑ j = 0 n { a j cos ⁡ j θ + b j sin ⁡ j θ } . The expected number of real zeros of T n ( θ ) in the interval ( 0 , 2 π ) can be easily obtained. In this note we show that this number is in fact n / 3 . However the variance of the above number is not known. This note presents a method which leads to the asymptotic value for the covariance of the number of real zeros of the above polynomial in intervals ( 0 , π ) and ( π , 2 π ) . It can be seen that our method in fact remains valid to obtain the result for any two disjoint intervals. The applicability of our method to the classical random trigonometric polynomial, defined as P n ( θ ) = ∑ j = 0 n a j ( ω ) cos ⁡ j θ , is also discussed. T n ( θ ) has the advantage on P n ( θ ) of being stationary, with respect to θ , for which, therefore, a more advanced method developed could be used to yield the results.