Let $\{X_t\}$ be a stationary time series and let $d_T(\lambda)$ denote the discrete Fourier transform (DFT) of $\{X_0,\ldots,X_{T-1}\}$ 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 $T(\lambda_{1T}- \lambda_{2T})=O(1)$ 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.
@article{1051027883,
author = {Lahiri, S. N.},
title = {A necessary and sufficient condition for asymptotic independence of discrete Fourier transforms under short- and long-range dependence},
journal = {Ann. Statist.},
volume = {31},
number = {1},
year = {2003},
pages = { 613-641},
language = {en},
url = {http://dml.mathdoc.fr/item/1051027883}
}
Lahiri, S. N. A necessary and sufficient condition for asymptotic independence of discrete Fourier transforms under short- and long-range dependence. Ann. Statist., Tome 31 (2003) no. 1, pp. 613-641. http://gdmltest.u-ga.fr/item/1051027883/