On Estimation of the Spectral Function of a Stationary Gaussian Process

Let x₁ ,x₂ , ⋯ ,xN be a sample time series drawn from the real stationary Gaussian process \ xₙ \ , Exₙ ≡ 0, with unknown spectral distribution function (s.d.f.) and spectral density function f(λ ). The problem of estimating of s.d.f. F(λ ) is discussed and the estimate FN^ * (λ ) = ∫₀^λ IN (λ )dλ of s.d.f. F(λ ) is considered, where\[ I_N (λ ) = {1}{{2π N}}| {∑_1^N {x_j eiλ j } } |^2 . \]In § $1 - 2$ the asymptotic properties of expressions like\[ { E}∫- π^π {φ (λ )I_N (λ )dλ } ,{ E}∫- π^π {T_1 (λ )I_N (λ )dλ } ∫- π^π {T_2 (μ )I_N (μ )dμ } \]are investigated. The main section of this paper is § 5. Let\[ ζ _N (λ ) = √ N [ {F_N^ * (λ ) - Fλ } ], \]and let ζ (λ ) be a Gaussian stochastic process with \[ ζ (0) = 0,Eζ (λ ) ≡ 0,{ E}ζ (λ )ζ (μ ) = 2π _0min (λ ,μ ) f^2 (λ )dλ , 0 λ ,μ π \]We denote by PN the probability measure induced in C[0,π ] by ζ N (λ ), and by P the probability measure induced in C[0,π ] by ζ (λ ). The following is proved in §5: Let1.∫ₐᵇ f(λ )dλ > 0 for every [a,b] ⊂ [ - π .π ];2.∫- π^π {f(λ )2 + δ dλ } < ∞ { for some }δ > 0;$then$ {PN ⇒ P}N → ∞ $, where the sign$ ⇒ $denotes weak convergence of the measures. In § 8 some estimates are given for probabilities of large deviations $F_N^ * (λ )$ from $F(λ )$. In § 9 it is shown that all results of §§ $1 - 8$ are valid for continuous time.