Apparent multifractality of self-similar L\'evy processes

TL;DR

This paper demonstrates that empirical moments can falsely suggest multifractality in self-similar Lévy processes, except for Brownian motion, due to their inherent scaling properties and the non-existence of certain moments.

Contribution

It reveals the limitations of empirical moments in correctly identifying monofractality in Lévy processes and introduces a stochastic normalization to address this issue.

Findings

Empirical moments can produce apparent multifractality in Lévy processes.

The piecewise-linear scaling function matches the stability index.

A stochastic normalization can correct the bias in empirical moments.

Abstract

Scaling properties of time series are usually studied in terms of the scaling laws of empirical moments, which are the time average estimates of moments of the dynamic variable. Nonlinearities in the scaling function of empirical moments are generally regarded as a sign of multifractality in the data. We show that, except for the Brownian motion, this method fails to disclose the correct monofractal nature of self-similar L\'evy processes. We prove that for this class of processes it produces apparent multifractality characterised by a piecewise-linear scaling function with two different regimes, which match at the stability index of the considered process. This result is motivated by previous numerical evidence. It is obtained by introducing an appropriate stochastic normalisation which is able to cure empirical moments, without hiding their dependence on time, when moments they aim at estimating do not exist.