Self-Similar Models: How close the diffusion entropy analysis and the detrended fluctuation analysis are from other models

Financial and seismic data, like many other high frequency data are known to exhibit memory effects. In this research, we apply the concepts of L ́evy processes, Diffusion Entropy Analysis (DEA) and the Detrended Fluctuation Analysis (DFA) to examine long-range persistence (long memory) behavior in time series data. L ́evy processes describe long memory effects. In other words, L ́evy process (where the increments are independent and follow the L ́evy distribution) is self-similar. We examine the relationship between the L ́evy parameter (α) characterizing the data and the scaling exponent of DEA (δ) and that of DFA (H) characterizing the self-similar property of the respective models. We investigate how close this model is to a self-similar model and prove the numerical relationship