Quantitative approach to multifractality induced by correlations and broad distribution of data

Abstract

We analyze quantitatively the effect of spurious multifractality induced by the presence of fat-tailed symmetric and asymmetric probability distributions of fluctuations in time series. In the presented approach different kinds of symmetric and asymmetric broad probability distributions of synthetic data are examined starting from Levy regime up to those with finite variance. We use nonextensive Tsallis statistics to construct all considered data in order to have good analytical description of frequencies of fluctuations in the whole range of their magnitude and simultaneously the full control over exponent of power-law decay for tails of probability distribution. The semi-analytical compact formulas are then provided to express the level of spurious multifractality generated by the presence of fat tails in terms of Tsallis parameter q̃ and the scaling exponent β of the asymptotic decay of cumulated probability density function (CDF). The results are presented in Hurst and Hölder languages — more often used in study of multifractal phenomena. According to the provided semi-analytical relations, it is argued how one can make a clear quantitative distinction for any real data between true multifractality caused by the presence of nonlinear correlations, spurious multifractality generated by fat-tailed shape of distributions — eventually with their asymmetry, and the correction due to linear autocorrelations in analyzed time series of finite length. In particular, the spurious multifractal effect of fat tails is found basic for proper quantitative estimation of all spurious multifractal effects. Examples from stock market data are presented to support these findings.