Abstract: | We consider a stationary process ( Xt , t = 0, ±1, ...) with a continuous spectrum. Denote by Dn (λ) a tapered Fourier transform of ( X 0, X 1, ..., X n −1) at (angular) frequency λ. We obtain the asymptotic distribution of Dn (λ) and the joint asymptotic distribution of { Dn (λ j ), 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←0 f (λ)λ2 d = K , a constant, where 0 < d < ½, as is typical of certain types of processes with long-range dependence. Results for the sample periodogram, proportional to | Dn (λ)|2, follow immediately. |