The level crossings of random polynomials (Q1921134)

The author obtains the expectation of the number of real zeros of the random trigonometric polynomial \(Q(\theta)=\sum_{j=1}^n a_j\sin j\theta\), \(0\leq \theta\leq 2\pi\), where the coefficients \(a_j\) are independent normally distributed random variables with mean 0 and variance 1 and proves that \(EN_0(0,2\pi)=2n/\sqrt 3+O(\sqrt n)\). With the same assumption for the coefficients of \(Q(\theta)\), the author proves that the expectation of the number of real roots of the equation \(Q(\theta)=k\) is given by \(EN_k(0,2\pi)=2n/\sqrt 3+O(\sqrt n+n^{2\alpha})\) where \(k_n=k=O(n^\alpha)\), \(0<\alpha<1/2\), \(\alpha\) being a constant.