Abstract-Multiplicative processes, multifractals and more recently also infinitely divisible cascades have seen increased popularity in a host of applications requiring versatile multiscale models, ranging from hydrodynamic turbulence to computer network traffic, from image processing to economics. The methodologies prevalent as of today rely to a large extent on iterative schemes used to produce infinite detail and repetitive structure across scales. While appealing due to their simplicity, these constructions have limited applicability as they lead by default to powerlaw progression of moments through scales, to nonstationary increments and often to inherent log-periodic scaling which favors an exponential set of scales. This article studies and develops on a wide class of infinitely divisible cascades (IDC), thereby establishing the first reported cases of controllable scaling of moments in non-powerlaw form. Embedded in the framework of IDC these processes exhibit stationary increments and scaling over a continuous range of scales. Criteria for convergence, further statistical properties as well as MATLAB routines are provided.

We study the statistical performance of multiresolution-based estimation procedures for the scaling exponents of multifractal processes. These estimators rely on the computation of multiresolution quantities such as wavelet, increment or aggregation coefficients. Estimates are obtained by linear fits performed in log of structure functions of order q versus log of scale plots. Using various and recent types of multiplicative cascades and a large variety of multifractal processes, we study and benchmark, by means of numerical simulations, the statistical performance of these estimation procedures. We show that they all undergo a systematic linearisation effect: for a range of orders q, the estimates account correctly for the scaling exponents; outside that range, the estimates significantly depart from the correct values and systematically behave as linear functions of q. The definition and characterisation of this effect are thoroughly studied. In contradiction with interpretations proposed in the literature, we provide numerical evidence leading to the conclusion that this linearisation effect is neither a finite size effect nor an infiniteness of moments effect, but that its origin should be related to the deep nature of the process itself. We comment on its importance and consequences for the practical analysis of the multifractal properties of empirical data.