Asymptotic Properties of the Detrended Fluctuation Analysis of Long-Range-Dependent Processes

In the past few years, a certain number of authors have proposed analysis methods of the time series built from a long-range dependence noise. One of these methods is the detrended fluctuation analysis (DFA), frequently used in the case of physiological data processing. The aim of this method is to highlight the long-range dependence of a time series with trend. In this paper, asymptotic properties of the DFA of the fractional Gaussian noise (FGN) are provided. Those results are also extended to a general class of stationary long-range-dependent processes. As a consequence, the convergence of the semiparametric estimator of the Hurst parameter is established. However, several simple examples also show that this method is not at all robust in the case of trends.