Linearization effect in multifractal analysis: Insights from the Random Energy Model

TL;DR

This paper investigates the linearization effect in multifractal analysis using concepts from the Random Energy Model, revealing a critical order beyond which moments cannot be reliably estimated, with implications for understanding multifractal exponents.

Contribution

It introduces a novel connection between multifractal analysis and the Random Energy Model, providing quantitative predictions for the critical order and linear behavior of multifractal exponents.

Findings

Existence of a critical order q* beyond which moments cannot be estimated.

Multifractal exponents behave linearly for q > q*.

Monte-Carlo simulations support the theoretical predictions.

Abstract

The analysis of the linearization effect in multifractal analysis, and hence of the estimation of moments for multifractal processes, is revisited borrowing concepts from the statistical physics of disordered systems, notably from the analysis of the so-called Random Energy Model. Considering a standard multifractal process (compound Poisson motion), chosen as a simple representative example, we show: i) the existence of a critical order $q^*$ beyond which moments, though finite, cannot be estimated through empirical averages, irrespective of the sample size of the observation; ii) that multifractal exponents necessarily behave linearly in $q$, for $q > q^*$. Tayloring the analysis conducted for the Random Energy Model to that of Compound Poisson motion, we provide explicative and quantitative predictions for the values of $q^*$ and for the slope controlling the linear behavior of the multifractal exponents. These quantities are shown to be related only to the definition of the multifractal process and not to depend on the sample size of the observation. Monte-Carlo simulations, conducted over a large number of large sample size realizations of compound Poisson motion, comfort and extend these analyses.