# EXPECTED NUMBER OF REAL ROOTS OF CERTAIN GAUSSIAN RANDOM TRIGONOMETRIC POLYNOMIALS

 Title EXPECTED NUMBER OF REAL ROOTS OF CERTAIN GAUSSIAN RANDOM TRIGONOMETRIC POLYNOMIALS Publication Type Journal Article Year of Publication 2016 Authors SHEMEHSAVAR SOUDABEH, FARAHMAND KAMBIZ Journal Neural, Parallel, and Scientific Computations Volume 24 Start Page 97 Pagination 10 Date Published 2016 ISSN 1061-5369 Abstract Let ${ D_n(θ) = \sum^n_{k=0}(A_k cos kθ + B_k sin kθ) }$ 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 ${ \frac{ 2 \sqrt{2}n}{\sqrt{3}}}$ . 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 ${ \frac{2n}{\sqrt{3} }.}$ URL https://acadsol.eu/npsc/articles/24/NPSC-97-106.pdf Refereed Designation Refereed