Journal article
EXPECTED NUMBER OF REAL ROOTS OF CERTAIN GAUSSIAN RANDOM TRIGONOMETRIC POLYNOMIALS
Neural, Parallel, and Scientific Computations , Vol.24, pp.97-106
2016
Abstract
Let D n (θ) = n k=0 (A k cos kθ + B k sin kθ) be a random trigonometric polynomial where the coefficients A 0 , A 1 ,. .. , A n , and B 0 , B 1 ,. .. , B n , form sequences of Gaussian random variables. Moreover, we assume that the increments ∆ 1 k = A k −A k−1 , ∆ 2 k = B k −B k−1 , k = 0, 1, 2,. .. , n, are independent, with conventional notation of A −1 = B −1 = 0. The coefficients A 0 , A 1 ,. .. , A n , and B 0 , B 1 ,. .. , B n , can be considered as n consecutive observations of a Brownian motion. In this paper we provide the asymptotic behavior of the expected number of real roots of D n (θ) = 0 as order 2 √ 2n √ 3. Also by the symmetric property assumption of coefficients, i.e., A k ≡ A n−k , B k ≡ B n−k , we show that the expected number of real roots is of order 2n √ 3 .
Details
- Title
- EXPECTED NUMBER OF REAL ROOTS OF CERTAIN GAUSSIAN RANDOM TRIGONOMETRIC POLYNOMIALS
- Authors/Creators
- Soudabeh Shemehsavar - Murdoch University, College of Science, Technology, Engineering and MathematicsFarahmand Kambiz
- Publication Details
- Neural, Parallel, and Scientific Computations , Vol.24, pp.97-106
- Publisher
- Dynamic Publishers Inc.
- Identifiers
- 991005728684907891
- Copyright
- © 2016 Dynamic Publishers Inc.
- Murdoch Affiliation
- College of Science, Technology, Engineering and Mathematics
- Language
- English
- Resource Type
- Journal article
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