Asymptotic normality of spectral estimates (Q1067335)

Let \(\{\) X(t)\(\}\), \(t\in {\mathbb{Z}}\), be a real valued weakly stationary time series with components \(X_ a(t)\), \(a=1,...,r\), mean 0, spectral density \(f(\lambda)=\{f_{ab}(\lambda)\}_{a,b=1,...,r}\). As an estimator of the spectral mean \(A_{ab}(\phi)=\int_{(-\pi,\pi]}\phi (\alpha)f_{ab}(\alpha)d\alpha\), \(\phi:(-\pi,\pi]\to C\) a function of bounded variation, the author considers \(A_{ab}^{(T)}(\phi)=\int_{(-\pi,\pi]}\phi (\alpha)I_{ab}^{(T)}(\alpha)d\alpha\) \((I_{ab}^{(T)}(\alpha)\) periodogram). Weak convergence of \[ \sqrt{T}(A^{(T)}_{a_ jb_ j}(\phi_ j)-EA^{(T)}_{a_ jb_ j}(\phi_ j)\}_{j=1,...,\ell}\quad and\quad \sqrt{T}(A^{(T)}_{a_ jb_ j}(\phi_ j)-A_{a_ jb_ j}(\phi_ j))_{j=1,...,\ell} \] to a complex-valued Gaussian random vector as well as in the space \(D^ k[0,1]\) with Skorokhod topology is considered. Using some recent results on the central limit theorem for stationary processes, corollaries are obtained for strong and \(\phi\)-mixing sequences and martingale differences. Related results were obtained earlier by \textit{A. V. Ivanov} and the reviewer, Litov. Mat. Sb. 18, No.4, 35-44 (1978; Zbl 0406.60024).