On the variance of the number of the number of real zeros of a random trigonometric polynomial

The asymptotic estimate of the expected number of real zeros of the polynomial T(θ)=g1cosθ+g2cos2θ+ëëë+gncosnθ where gj (j=1,2,...,n) is a sequence of independent normally distributed random variables is known. The present paper provides an upper estimate for the variance of such a number. To achieve this result the paper first presents a general formula for the covariance of the number of real zeros of any normal process, ξ(t), µoccurring in any two disjoint intervals. A formula for the variance of the number of real zeros of ξ(t) follows from this result.