Here we apply different methods to establish the Gaussian approximation to linear statistics of a stationary sequence, including stationary linear processes, near-stationary processes, and discrete Fourier transforms of a strictly stationary process. More precisely, we analyze the asymptotic behavior of the partial sums associated with a short-memory linear process and prove, in particular, that if a weak limit theorem holds for the partial sums of the innovations then a related result holds for the partial sums of the linear process itself. We then move to linear processes with long memory and obtain the CLT under various dependence structures for the innovations by analyzing the asymptotic behavior of linear statistics. We also deal with the invariance principle for causal linear processes or for linear statistics with weakly associated innovations. The last section deals with discrete Fourier transforms, proving, via martingale approximation, central limit behavior at almost all frequencies under almost no condition except a regularity assumption.
Fluctuations of Linear Statistics for Gaussian Perturbations of the Lattice $${\mathbb {Z}}^d$$