Hausdorff, Large Deviation and Legendre Multifractal Spectra of L\'evy Multistable Processes

TL;DR

This paper analyzes the multifractal spectra of multistable Lévy processes, revealing differences between Hausdorff and Legendre spectra and providing insights into the multifractal formalism's limitations.

Contribution

It computes the Hausdorff and large deviation multifractal spectra of multistable Lévy motions, highlighting cases where the multifractal formalism fails.

Findings

Hausdorff spectrum decomposes [0,1] into sets of Hausdorff dimension one.

Large deviation spectrum is concave and matches the Legendre spectrum.

The independent increments multistable Lévy motion exemplifies the failure of the strong multifractal formalism.

Abstract

We compute the Hausdorff multifractal spectrum of two versions of multistable L{\'e}vy motions. These processes extend classical L{\'e}vy motion by letting the stability exponent $\alpha$ evolve in time. The spectra provide a decomposition of [0, 1] into an uncountable disjoint union of sets with Hausdorff dimension one. We also compute the increments-based large deviations multifractal spectrum of the independent in-crements multistable L{\'e}vy motion. This spectrum turns out to be concave and thus coincides with the Legendre multifractal spectrum, but it is different from the Haus-dorff multifractal spectrum. The independent increments multistable L{\'e}vy motion thus provides an example where the strong multifractal formalism does not hold.