An Example of non-quenched Convergence in the Conditional CLT for Discrete Fourier Transforms

A recent result by Barrera and Peligrad shows that the quenched Central Limit Theorem holds for the discrete Fourier transforms (DFT) of a stationary process in $L^2$ if a “random” centering is used. In this note we show that this is a necessary condition by providing an example of a process satisfying the hypothesis of such theorem for which the DFT, without random centering, do not satisfy a quenched limit theorem. The DFT of this process, by previous results by Peligrad and Wu, satisfy the corresponding CLT in the annealed sense.