Multifractal analysis of self-similar processes

Scale invariance and multifractal analysis are nowadays widely used in applications. For modeling scale invariance in data, two classes of processes are classically in competition: self-similar processes and multiplicative cascades. They imply fundamentally different underlying (additive or multiplicative) mechanisms, hence the crucial practical need for data driven model selection. Such identification relies on properties often associated with the former: self-similarity, monofractality, linear scaling function, null c₂ parameter. By performing a wavelet leader based analysis of the multifractal properties of a large variety of self-similar processes, the present work contributes to a better disentangling of these different properties, sometimes confused one with another. Also, it leads to the formulation of conjectures regarding the scaling and multifractal properties of self-similar processes.