Some Results Concerning the Asymptotic Distribution of Sample Fourier Transforms and Periodograms for a Discrete‐time Stationary Process with a Continuous Spectrum

A. M. Walker

Journal of Time Series Analysis, 2000, vol. 21, issue 1, 95-109

Abstract: We consider a stationary process (Xt, t = 0, ±1, ...) with a continuous spectrum. Denote by Dn(λ) a tapered Fourier transform of (X0, X1, ..., Xn−1) at (angular) frequency λ. We obtain the asymptotic distribution of Dn(λ) and the joint asymptotic distribution of {Dn(λj), 1 ≤j≤k} with continuity of the spectral density f(.) at the relevant frequencies as the only assumption concerning the second‐order structure of (Xt); all other assumptions required are easily stated. The results are extended to processes for which f(.) is continuous except at λ = 0, with limλ←0f(λ)λ2d = K, a constant, where 0

Date: 2000
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