Journal of Applied Mathematics and Stochastic Analysis
Volume 10 (1997), Issue 1, Pages 57-66

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

K. Farahmand

University of Ulster, Co. Antrim , Jordanstown BT37 0QB, United Kingdom

Received 1 March 1994; Revised 1 September 1995

Copyright © 1997 K. Farahmand. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The asymptotic estimate of the expected number of real zeros of the polynomial T(θ)=g1cosθ+g2cos2θ++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 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 of ξ(t) follows from this result.