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.