Journal of Applied Mathematics and Stochastic Analysis
Volume 8 (1995), Issue 3, Pages 299-317

Mean number of real zeros of a random trigonometric polynomial. III

J. Ernest Wilkins Jr. and Shantay A. Souter

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.


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.