On the variance of the number of real roots of a random trigonometric polynomial (Q751727)

Let \(T(\theta)=g_ 1 \cos \theta +g_ 2 \cos 2\theta +...+g_ n \cos n\theta\) be a random trigonometric polynomial, where the coefficients \(g_ i\) \((i=1,2,...,n)\) are assumed to be independent standard normal random variables. Then it is proved that the variance of the number of real zeros of T(\(\theta\)) in the interval (0,2\(\pi\)) is \(O(n^{24/13}(\log n)^{16/13})\). It is remarked that this result is also true for dependent random coefficients with mean zero or nonzero.