Lahiri, S. N. A necessary and sufficient condition for asymptotic independence of discrete Fourier transforms under short- and long-range dependence. (English) Zbl 1039.62087 Ann. Stat. 31, No. 2, 613-641 (2003). 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. Cited in 24 Documents MSC: 62M15 Inference from stochastic processes and spectral analysis 62E20 Asymptotic distribution theory in statistics 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) Keywords:asymptotic independence; discrete Fourier transform; long-range dependence; stationarity Software:longmemo × Cite Format Result Cite Review PDF Full Text: DOI References: [1] ADENSTEDT, R. K. (1974). On large-sample estimation for the mean of a stationary random sequence. Ann. Statist. 2 1095-1107. · Zbl 0296.62081 · doi:10.1214/aos/1176342867 [2] BERAN, J. (1994). Statistics for Long-Memory Processes. Chapman and Hall, London. · Zbl 0869.60045 [3] BRILLINGER, D. (1981). Time Series: Data Analy sis and Theory, 2nd ed. Holden-Day, San Francisco. [4] BROCKWELL, P. J. and DAVIS, R. A. (1991). Time Series: Theory and Methods, 2nd ed. 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