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=2½{(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 105 when n8, and by only about 0.1% when n=2.