Generalized (m,k)-Zipf law for fractional Brownian motion-like time series with or without effect of an additional linear trend

Abstract

We have translated fractional Brownian motion (FBM) signals into a text based on two &#39;&#39;letters&#39;&#39;, as if the signal fluctuations correspond to a constant stepsize random walk. We have applied the Zipf method to extract the $ζ&#39;$ 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: $ζ&#39;$ is varying as a power law in terms of $m$. We have also searched how the $ζ&#39;$ 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 $ζ&#39; = |2H-1|$ only holds near $H=0.5$. We have also introduced considerations based on the notion of a {\it time dependent Zipf law} along the signal.