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

Let {X_t} be a stationary time series and let d _T(λ) denote the discrete Fourier transform (DFT) of {X ₀, ..., 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(λ_1T) and d _T(λ_2T) are also established for the cases T(λ_1T - λ_2T) = O(1) as T → ∞ (asymptotically close ordinates) and |T(λ_1T - λ _2T)| → ∞ as T → ∞ (asymptotically distant ordinates). Some implications of the main results on the estimation of the index of dependence are also discussed.