# "Central limit theorem" for weighted sum of correlated random variables

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.

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.

• What are those conditions? And how does his theorem differ from the paper I cite? Aug 13, 2012 at 21:46
• It is probably very similar to the result in the paper that you cite. I looked it up because it sounded something like a result i learned back in my graduate school days. I am not going to recite the assumptions. It involves a constraint on the autocorrelation function for Xj and the λjs don't sum in pairs to multiples of 2π. Aug 13, 2012 at 22:13