Spectral density estimation for linear processes with dependent innovations

This paper considers the problem of estimating the spectral density of a linear process whose innovations are uncorrelated and strongly mixed. We prove that the Periodogram ordinates In(λi)In(λi) at any set of frequencies λ1,…,λm,0<λ1<⋯<λm<πλ1,…,λm,0<λ1<⋯<λm<π, are asymptotically independent exponential random variables with means 2πf(λi)2πf(λi). Consequently the periodogram InIn is not a consistent estimator of 2πf2πf. Consistent estimators can, however, be constructed by applying linear smoothing filters to the periodogram.