A necessary and sufficient condition for asymptotic independence of discrete Fourier transforms under short- and long-range dependence. (English) Zbl 1039.62087

Summary: Let \(\{X_t\}\) be a stationary time series and let \(d_T (\lambda)\) denote the discrete Fourier transform (DFT) of \(\{X_0,\dots, X_{T-1}\}\) with a data taper. The main results of this paper provide a characterization of asymptotic independence of the DFTs in terms of the distance between their arguments under both short- and long-range dependence of the process \(\{X_t\}\). Further, asymptotic joint distributions of the DFTs \(d_T(\lambda_1T)\) and \(d_T (\lambda_{2T})\) are also established for the cases \(T(\lambda_{1T}- \lambda_{2 T})= O (1)\) as \(T\to\infty\) (asymptotically close ordinates) and \(| T( \lambda_{ 1_T}-\lambda_{2_T}) |\to\infty\) as \(T\to\infty\) (asymptotically distant ordinates). Some implications of the main results on the estimation of the index of dependence are also discussed.


62M15 Inference from stochastic processes and spectral analysis
62E20 Asymptotic distribution theory in statistics
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)


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