2002/09/13 by Ph. Bronlet, Marcel Ausloos, M. Ausloos · 1 citation
Computer Science · Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Artificial intelligence #Brownian motion #Complex Systems and Time Series Analysis #Computer science #Financial Risk and Volatility Modeling #Font #Fractional Brownian motion #Geology #Mathematics #Nonlinear Dynamics and Pattern Formation #Physics #Series (stratigraphy) #Statistical physics #Statistics #Zipf's law #cond-mat
paper · pdf · doi:10.1142/s0129183103004528
24 pages, 12 figures; to appear in Int. J. Modern Phys C
arxiv created 2002/09/13 · openalex publication_date 2003/03/01 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We have translated fractional Brownian motion (FBM) signals into a text based on two "letters", as if the signal fluctuations correspond to a constant stepsize random walk. We have applied the Zipf method to extract the ζ′ exponent relating the word frequency and its rank on a log–log plot. We have studied the variation of the Zipf exponent(s) giving the relationship between the frequency of occurrence of words of length m < 8 made of such two letters: ζ′ is varying as a power law in terms of m. We have also searched how the ζ′ exponent of the Zipf law is influenced by a linear trend and the resulting effect of its slope. We can distinguish finite size effects, and results depending whether the starting FBM is persistent or not, i.e., depending on the FBM Hurst exponent H. It seems then numerically proven that the Zipf exponent of a persistent signal is more influenced by the trend than that of an antipersistent signal. It appears that the conjectured law ζ′ = |2H - 1| only holds near H = 0.5. We have also introduced considerations based on the notion of a time dependent Zipf law along the signal.