EXPECTED NUMBER OF REAL ROOTS OF CERTAIN GAUSSIAN RANDOM TRIGONOMETRIC POLYNOMIALS

摘要

Let D_n(θ) = Σ_(k=0)~n(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 Δ_k~1 = A_k-A_(k-1), Δ_k~2 = 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 2~(1/2)n)/3~(1/2). 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~(1/2).