Long-range correlation properties of stationary linear models with mixed periodicities.

We consider the problem of (stationary and linear) source systems which generate time series data with long-range correlations. We use the discrete Fourier transform (DFT) and build stationary linear models using artificial time series data exhibiting a 1/f spectrum, where the models can include only terms that contribute significantly to the model as assessed by information criteria. The result is that the optimal (best) model is only composed of mixed periodicities [that is, the model does not include all (continuous) periodicities] and the time series data generated by the model exhibit a clear 1/f spectrum in a wide frequency range. It is considered that as the 1/f spectrum is a consequence of the contributions of all periods, consecutive periods are indispensable to generate such data by stationary linear models. However, the results indicate that there are cases where this expectation is not always met. These results also imply that although we can know linear features of time series data using the DFT, we always cannot substantially infer the type of the source system, even if the system is stationary linear.