Multifractality attributed to dual central limit-like convergence effects

Abstract

Multifractals can be defined as fractal systems that express a range of fractal dimensions. The origins of multifractality in time series data have conventionally been attributed to fat-tailed probability distributions, and to long-range correlations. Multifractal sequences can be generated from the eigenvalue deviations of the Gaussian unitary and orthogonal ensembles of random matrix theory. These deviations can be resolved into component monofractal sequences governed by the Tweedie compound Poisson distribution, a statistical model that expresses a variance to mean power law related to long-range correlations. Fully multifractal descriptions of these deviations can be constructed, provided that the parameter of the compound Poisson model related to fractal dimension varies in accordance with an asymmetric Laplace distribution. Both the Tweedie compound Poisson distribution and the asymmetric Laplace distribution serve as foci of convergence in limit theorems on independent and identically distributed random variables. The hypothesis that multifractal sequences can be attributed to mathematical convergence effects that have as their focus these two statistical models is proposed.