Journal of Applied Mathematics and Stochastic Analysis
Volume 10 (1997), Issue 1, Pages 57-66
doi:10.1155/S1048953397000051
On the variance of the number of real zeros of a random trigonometric polynomial
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.
Abstract
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.