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

[摘要] The asymptotic estimate of the expected number of real zeros of the polynomialT(θ)=g1cosθ+g2cos2θ+…+gncosnθwheregj(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 we first present a general formula for the covariance of the number of real zeros of any normal process,ξ(t), occurring in any two disjoint intervals. A formula for the variance of the number of real zeros ofξ(t)follows from this result.

[学科分类] 应用数学