Journal of Applied Mathematics and Stochastic Analysis
Volume 8 (1995), Issue 3, Pages 299-317
doi:10.1155/S104895339500027X
Mean number of real zeros of a random trigonometric polynomial. III
Clark Atlanta University, Department of Mathematical Sciences, Atlanta 30314, GA, USA
Received 1 August 1994; Revised 1 March 1995
Copyright © 1995 J. Ernest Wilkins and Shantay A. Souter. 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
If a1,a2,…,an are independent, normally distributed random variables with
mean 0 and variance 1, and if vn is the mean number of zeros on the interval
(0,2π) of the trigonometric polynomial a1cosx+2½a2cos2x+…+n½ancosnx,
then vn=2−½{(2n+1)+D1+(2n+1)−1D2+(2n+1)−2D3}+O{(2n+1)−3}, in which D1=−0.378124, D2=−12, D3=0.5523. After tabulation of
5D values of vn when n=1(1)40, we find that the approximate formula for vn,
obtained from the above result when the error term is neglected, produces 5D
values that are in error by at most 10−5 when n≥8, and by only about 0.1%
when n=2.