CONVERGENCE RATES OF ESTIMATE OF SPECTRAL FUNCTION FOR STATIONARY GAUSSIAN SERIES

Suppose X=x（n）, n=0, ±1, … is a real strictly stationary process, Ex（n）=0, Ex（n）x（m）=B（n-m）.Let f（λ） and F \ddot( )=\int_0^\lambda f(u) d u be the spectral density and spectral function of X respectively. One way to estimate B（k） and F（λ） from N consecutive observations x（j）,j= 1, 2, …,N is by means of \begingathered \hatB_N(k)= \begincases\frac1N \sum_n=1^N-k x\left(n^\prime x(n+k)\right. & 0 \leqslant k \leqslant N-1 \\ 0 & k \geqslant N\endcases \\ \hatF_N(\lambda)=\int_0^\lambda \frac12 \pi N \sum_k=1^X x(k) e^-i k u y^2 d u \endgathered and respectively. This paper is mainly concerned with the a. s. convergence rates of |\hatF_N(\lambda)-F(\lambda)| and \sup _0<\lambda x\left|\hatF_N(\lambda)-F(\lambda)\right| for a real stationary Gaussian process. First, we establish the a. s. convergence rates of the quadratic forms of the observations. Then we obtain the convergence rates of estimates of the covariances and spectral function. The main results are as follows. Lemma 1. Let X= x（n） be a real stationary Gaussian process with zero mean and B_N= （B （n-m）） the N×N covariance matrix of X. Y_N= X’_NA_NX_N-EX’_NA_NX_N where X’_N=（x（1）, …, x（N））,A_N is a N×N real symmetric matrix. If λ_μ is the maximum of the absolute value of the eigenvalues of B_N^1/2A_NB_N^1/2, then for any fixed δ＞0 and 0≤α≤δ/2（1+δ）λ_μ^-1, we have \begingatheredE\lefte^\alpha Y_N\right \leqslant \exp \left\\frac1+\delta2 \alpha^2 \operatornameVar\left(Y_N\right)\right\ \\ \text and \quad E\lefte^\alpha \mid Y_X i\right \leqslant 2 \exp \left\\frac1+\delta2 \alpha^2 \operatornameVar\left(Y_N\right)\right\ .\endgathered. Theorem 1. Let X=\x(n)\ be a real stationary Gaussian process with zero mean, and f^2 \log ^+ f \in L0, \pi. Y_N=X_N^\prime A_N X_N-E X_N^\prime \cdot A_N X_N where X_N^\prime=(x(1), \cdots, x(N)), A_N is a N \times N real symmetric matrix. If \left\|A_*\right\| \triangle \sup _1 \mathrmI_2=1\left|Y^\prime A_N Y\right| \leqslant c \quad N \geqslant 1 and \lim _N \rightarrow \infty \frac1N \operatornameVar\left(Y_N\right)=a^2>0 'I'hen \varlimsup_N \rightarrow \infty \frac1\sqrt2 \overlineN \log N\left|Y_N\right| \leqslant a \quad \text a.s. Corollary 1. Let X=\x(n)\ satisfy the conditions of Theorem 1. Then (1) \forall k \geqslant 0 \left.\varlimsup_N \rightarrow \infty \sqrt\fracN2 \log N \hatB_N(k)-B(k) \right\rvert\, \leqslant d_k \quad \text a.s. where d_k^2=4 \pi \int_-\pi^\pi \cos ^2 k u f^2(u) d u (2) \forall \lambda \in0, \pi \varlimsup_N \rightarrow \infty \sqrt\fracN2 \log N\left|\hatF_N(\lambda)-F(\lambda)\right| \leqslant c_\lambda \quad \text a.s. where c_\lambda^2=2 \pi \int_0^\lambda f^2(u) d u Theorem 2. Let X=\x(n)\ satisfy the conditions of Theorem 1. Then for \begingathered P(N)=O(\sqrtN \log N) \\ \varlimsup_N \rightarrow \infty \sqrt\fracN2 \log N \sup _0< k< P(N)\left|\hatB_N(k)-B(k)\right| \leqslant \sqrt\frac32 d_0 \quad \text a.s. \\ d_0^2=4 \pi \int_-x^\pi f^2(u) d u . \endgathered where Moreover, if f \in \operatornameLip-\frac12 in 0, \pi, then \varlimsup_N \rightarrow \infty \sqrt\fracN2 \log N \sup _0< k< \infty\left|\hatB_N(k)-B(k)\right| \leqslant \sqrt2 d_0 \quad \text a.s. Theorem 3. Let X=\x(n)\ satisfy the conditions of Theorem 1. Then \varlimsup_N \rightarrow \infty \sqrt\fracN2 \log N \sup _0< \lambda< x\left|\hatF_N(\lambda)-F(\lambda)\right| \leqslant(\sqrt2+2) c_x \quad \text a.s. where c_\pi^2=2 \pi \int_0^\pi f^2(u) d u . Finally, using a result of 5, we obtain the convergence rates of estimate of spectral function for a real linear process. Proposition. Let X=\x(n)\ be a real linear process, x(n)=\sum_j=0^\infty \alpha(j) \varepsilon(n-j) \quad \sum_j=0^\infty|\alpha(j)|< \infty \quad \alpha(0)=1 where \\varepsilon(n)\ is a strictly stationary martingale difference and E\left(\varepsilon^2(n) \mid \mathscrF_n-1\right)=\sigma^2, \quad E \varepsilon^4(n)< \infty ; \mathscrF_n \triangleq \sigma\\varepsilon(m) ; m \leqslant n\ . If Then \begingathered \lim _N \rightarrow \infty \sqrtN \sum_k=N^\infty|\alpha(k)|=0 \\ \sup _0<\lambda<=\left|\hatF_N(\lambda)-F(\lambda)\right|=O\left(N^-\frac12(\log N)^\frac32\right) \quad \text as \endgathered