GENERALIZED (m, k)-ZipfLAW 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 "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 loglog 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.

Stock index futures trading impact on spot price volatility. The CSI 300 studied with a TGARCH model

“…Thus, it is concluded that the sample of the daily returns has significant autocorrelation and partial autocorrelation at lag 4. It can be deduced that such a finding indicates that the stock price can be predicted on such a time span (Ausloos and Bronlet, 2003;Bronlet and Ausloos, 2003).…”

A Volatility Estimator of Stock Market Indices Based on the Intrinsic Entropy Model

“…In Appendix D, we provide Figures A6-A10, which show the evolution of Yang-Zhang and intrinsic entropy-based estimates for Dow 30, Russel 2000, NASDAQ Composite, Nikkei 225, and Hang Seng stock market indices, respectively, over a time window of 260 days, going backwards from 31 January 2021. From this different perspective of market volatility, we note that the intrinsic entropy-based volatility estimates may have a more useful role in emphasizing the fractured nature of the market [28,29]. Robert A.…”

“…Perhaps this observation of "relatively strong price movements" has something to do with the interest that the investors show in those securities, something which may be due to the fundamentals of those securities. The intrinsic value of these securities appears to From this different perspective of market volatility, we note that the intrinsic entropybased volatility estimates may have a more useful role in emphasizing the fractured nature of the market [28,29]. Robert A.…”

Econophysics: Comments on a Few Applications, Successes, Methods and Models

“…Another sort of data analysis technique can be adapted to portfolio management (Ausloos & Bronlet, 2006), leading to forecasting and prediction, with some ‘risk’: it is known as the Zipf technique, originating in work exploring the statistical nature of languages (Zipf, 1949); see Appendix 3. The Zipf method previously developed as an investment strategy (on usual financial indices) (Ausloos & Bronlet, 2003; Bronlet & Ausloos, 2003) can be adapted to portfolio management.…”