Multifractal analysis and α-stable processes: a methodological contribution

This work is a contribution to the analysis of the procedure, based on wavelet coefficient partition functions, commonly used to estimate the Legendre multifractal spectrum. The procedure is applied to two examples, a fractional Brownian motion in multifractal time and a self-similar α-stable process, whose sample paths exhibit irregularities that by eye appear very close. We observe that, for the second example, this analysis results in a qualitatively inaccurate estimation of its multifractal spectrum, and a related masking of the α-stable nature of the process. We explain the origin of this error through a detailed analysis of the partition functions of the self-similar α-stable process. Such a study is made possible by the specific properties of the wavelet coefficients of such processes. We indicate how the estimation procedure might be modified to avoid such errors