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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Mean number of real zeros of a random trigonometric polynomial


Author: J. Ernest Wilkins
Journal: Proc. Amer. Math. Soc. 111 (1991), 851-863
MSC: Primary 60G99
MathSciNet review: 1039266
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Abstract: If $ {a_1},{a_2}, \ldots ,{a_n}$ are independent, normally distributed random variables with mean 0 and variance 1, and if $ {\nu_n}$ is the mean value of the number of zeros on the interval $ (0,2\pi )$ of the trigonometric polynomial $ {a_1}\cos x + {a_2}\cos 2x + \cdots + {a_n}\cos nx$, then

$\displaystyle {\nu_n} = {3^{1/2}}\{ (2n + 1) + {D_1} + {(2n + 1)^{ - 1}}{D_2} + {(2n + 1)^{ - 2}}{D_3}\} + O\{ {(2n + 1)^{ - 3}}\} ,$

in which $ {D_1} = 0.232423 \cdots ,{D_2} = - 0.25973 \cdots$, and $ {D_3} = 0.2172 \cdots $.

References [Enhancements On Off] (What's this?)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1039266-0
PII: S 0002-9939(1991)1039266-0
Article copyright: © Copyright 1991 American Mathematical Society