Kammoun I: Asymptotic properties of the detrended fluctuation analysis of long-range-dependent processes

Abstract — 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 study asymptotic properties of DFA of the fractional Gaussian noise are provided. Those results are also extended to a general class of stationary long-range dependent processes. As a consequence, the convergence of the semi-parametric estimator of the Hurst parameter is established. However, several simple exemples also show that this method is not at all robust in case of trend. Index Terms — Detrended ¤uctuation analysis, fractional Gaus-sian noise, stationary process, self-similar process, Hurst param