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

J. Ernest Wilkins and Shantay A. Souter

International Journal of Stochastic Analysis, 1995, vol. 8, 1-19

Abstract:

If a 1 , a 2 , … , a n are independent, normally distributed random variables with mean 0 and variance 1 , and if v n is the mean number of zeros on the interval ( 0 , 2 π ) of the trigonometric polynomial a 1 cos x + 2 ½ a 2 cos 2 x + … + n ½ a n cos n x , then v n = 2 − ½ { ( 2 n + 1 ) + D 1 + ( 2 n + 1 ) − 1 D 2 + ( 2 n + 1 ) − 2 D 3 } + O { ( 2 n + 1 ) − 3 } , in which D 1 = − 0.378124 , D 2 = − 1 2 , D 3 = 0.5523 . After tabulation of 5 D values of v n when n = 1 ( 1 ) 40 , we find that the approximate formula for v n , obtained from the above result when the error term is neglected, produces 5 D values that are in error by at most 10 − 5 when n ≥ 8 , and by only about 0.1 % when n = 2 .

Date: 1995
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnijsa:950891

DOI: 10.1155/S104895339500027X

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