EXPECTED NUMBER OF REAL ROOTS OF CERTAIN GAUSSIAN RANDOM TRIGONOMETRIC POLYNOMIALS

TitleEXPECTED NUMBER OF REAL ROOTS OF CERTAIN GAUSSIAN RANDOM TRIGONOMETRIC POLYNOMIALS
Publication TypeJournal Article
Year of Publication2016
AuthorsSHEMEHSAVAR SOUDABEH, FARAHMAND KAMBIZ
JournalNeural, Parallel, and Scientific Computations
Volume24
Start Page97
Pagination10
Date Published2016
ISSN1061-5369
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 ${ \frac{2n}{\sqrt{3} }.}$

URLhttps://acadsol.eu/npsc/articles/24/NPSC-97-106.pdf
Refereed DesignationRefereed