Mean number of real zeros of a random trigonometric polynomial. IIIReport as inadecuate




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

Clark Atlanta University, Department of Mathematical Sciences, Atlanta 30314, GA, USA

Received 1 August 1994; Revised 1 March 1995

Copyright © 1995 Hindawi Publishing Corporation. 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=1140, 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.





Author: J. Ernest Wilkins Jr. and Shantay A. Souter

Source: https://www.hindawi.com/



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