## The Annals of Statistics

### A necessary and sufficient condition for asymptotic independence of discrete Fourier transforms under short- and long-range dependence

S. N. Lahiri

#### Abstract

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 information

Source
Ann. Statist., Volume 31, Number 2 (2003), 613-641.

Dates
First available in Project Euclid: 22 April 2003

https://projecteuclid.org/euclid.aos/1051027883

Digital Object Identifier
doi:10.1214/aos/1051027883

Mathematical Reviews number (MathSciNet)
MR1983544

Zentralblatt MATH identifier
1039.62087

#### Citation

Lahiri, S. N. A necessary and sufficient condition for asymptotic independence of discrete Fourier transforms under short- and long-range dependence. Ann. Statist. 31 (2003), no. 2, 613--641. doi:10.1214/aos/1051027883. https://projecteuclid.org/euclid.aos/1051027883

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• AMES, IOWA 50011-1210 E-MAIL: snlahiri@iastate.edu