EXPECTED NUMBER OF REAL ROOTS OF CERTAIN GAUSSIAN RANDOM TRIGONOMETRIC POLYNOMIALS

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 ${ \frac{2n}{\sqrt{3} }.}$