“Central limit theorem” for weighted sum of correlated random variables

Question

I'm reading a paper which claims that

$$\hat{X}_k=\frac{1}{\sqrt{N}}\sum_{j=0}^{N-1}X_je^{-i2\pi kj/N},$$ (i.e. the Discrete Fourier Transform, DFT) by the C.L.T. tends to a (complex) gaussian random variable. However, I know this isn't true in general. After reading this (fallacious) argument, I searched over the net and found this 2010 paper by Peligrad & Wu, where they prove that for some stationary processes, one can find a "C.L.T. theorem".

My question is: do you have any other references that try to address the problem of finding the limiting distribution of the DFT of a given indexed sequence (both by simulation or theory)? I'm particularly interested in the convergence rate (i.e. how quickly the DFT converges) given some covariance structure for $X_j$ in the context of time-series analysis, or derivations/applications to non-stationary series.

Answer 1

In David Brillinger's "Time Series Data Analysis and Theory" 1975 Holt, Rinehart and Winston Publishers page 94 Theroem 4.4.1 states under certain condition the discrete fourier transform for an r vector-valued series at frequencies λ$_j$(N) are asymptotically independent r dimensional complex normal variates with mean vector 0 where λ$_j$(N)=2π s$_j$(N)/N. This happens to be a very important theorem in the development of estimates for the spectral density of stationary time series.