Hausdorff, large deviation and Legendre multifractal spectra of Lévy multistable processes

Abstract

We compute the Hausdorff multifractal spectrum of two versions of multistable Lévy motions. These processes extend classical Lévy motion by letting the stability exponent α evolve in time. The spectrum provides 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 increments multistable Lévy motion. This spectrum turns out to be concave and thus coincides with the Legendre multifractal spectrum, but it is different from the Hausdorff multifractal spectrum. The independent increments multistable Lévy motion thus provides an example where the strong multifractal formalism does not hold.