The level crossings of random polynomials

The present paper investigates the properties of the expected number of real zeros and K-level crossings of random trigonometric polynomials a(1) sin theta + a(2) sin 2 theta +...+ a(n) sin n theta, where a(j), j = 1, 2,..., n are independent normally distributed random variables. It is shown that the result for K = 0 remains valid for any K such that K = O(n(a)) where 0 less than or equal to alpha < 1/2 is a constant.