On Estimation of the Spectral Function of a Stationary Gaussian Process

Let $x_1,x_2,\dots,x_N$ be a sample time series drawn from the real stationary Gaussian process $\{x_n\},{\mathbf E}x_n\equiv0$, with unknown spectral distribution function (s.d.f.) and spectral density function $f(\lambda)$. The problem of estimating of s.d.f. $F(\lambda)$ is discussed and the estimate $F_N^*(\lambda)=\int_0^\lambda{I_N} (\lambda )d\lambda$ of s.d.f. $F(\lambda)$ is considered, where
$$I_N(\lambda)=\frac{1}{{2\pi N}}\left| {\sum\limits_1^N{x_j e^{i\lambda j}}}\right|^2.$$
In §1–§2 the asymptotic properties of expressions like
$${\mathbf E}\int_{-\pi}^\pi{\varphi(\lambda)I_N(\lambda)\,d\lambda},\quad{\mathbf E}\int_{-\pi}^\pi{T_1(\lambda)I_N(\lambda )\,d\lambda}\int_{-\pi}^\pi{T_2(\mu )I_N(\mu)\,d\mu}$$
are investigated. The main section of this paper is §5. Let
$$\zeta_N(\lambda)=\sqrt N\left[{F_N^*(\lambda)-F\lambda}\right],$$
and let $\zeta(\lambda)$ be a Gaussian stochastic process with
$$\zeta(0)=0,\,\mathbf E\zeta(\lambda)\equiv0,\,{\mathbf E}\zeta(\lambda)\zeta(\mu)=2\pi\displaystyle\int_0^{\min(\lambda,\mu)}f^2(\lambda)\,d\lambda,\quad0\leq\lambda,\,\mu\leq\pi.$$
We denote by $P_N$ the probability measure induced in $C[0,\pi]$ by $\zeta_N(\lambda)$, and by $P$ the probability measure induced in $C[0,\pi]$ by $\zeta(\lambda)$. The following is proved in §5:
Theorem 5.1 Let
$$1.\int_a^b{f(\lambda)\,d\lambda}>0\qquad{\text{for every}}\quad[a,b]\subset[-\pi,\pi];\\ 2.\int_{-\pi}^\pi{(f(\lambda))^{2+\delta}\,d\lambda}<\infty\qquad{\text{for some}}\quad\delta>0,$$
then $\mathop{P_N\Rightarrow P}\limits_{N\to\infty}$, where the sign $\Rightarrow$ denotes weak convergence of the measures.
In §8 some estimates are given for probabilities of large deviations $F_N^*(\lambda)$ from $F(\lambda)$.
In §9 it is shown that all results of §§$1$–$8$ are valid for continuous time.