Generic Multifractality in Exponentials of Long Memory Processes

TL;DR

This paper demonstrates that multifractal scaling is a robust property of exponential long-memory processes, with tunable intermittency coefficients, broad scaling regimes, and universal spectral collapse, extending previous models with zero memory tail exponent.

Contribution

It generalizes previous multifractal models by incorporating a positive power law tail exponent, showing multifractality over a wide scale range, and revealing a universal scaling function for multifractal spectra.

Findings

Multifractality persists over large scale ranges for processes with positive tail exponent.

The multifractal spectrum can be collapsed onto a universal scaling function.

High-order multifractal exponents can be derived from small-order values.

Abstract

We find that multifractal scaling is a robust property of a large class of continuous stochastic processes, constructed as exponentials of long-memory processes. The long memory is characterized by a power law kernel with tail exponent $\phi+1/2$, where $\phi >0$. This generalizes previous studies performed only with $\phi=0$ (with a truncation at an integral scale), by showing that multifractality holds over a remarkably large range of dimensionless scales for $\phi>0$. The intermittency multifractal coefficient can be tuned continuously as a function of the deviation $\phi$ from 1/2 and of another parameter $\sigma^2$ embodying information on the short-range amplitude of the memory kernel, the ultra-violet cut-off (``viscous'') scale and the variance of the white-noise innovations. In these processes, both a viscous scale and an integral scale naturally appear, bracketing the ``inertial'' scaling regime. We exhibit a surprisingly good collapse of the multifractal spectra $\zeta(q)$ on a universal scaling function, which enables us to derive high-order multifractal exponents from the small-order values and also obtain a given multifractal spectrum $\zeta(q)$ by different combinations of $\phi$ and $\sigma^2$.