Extreme Values, Heavy Tails and Linearization Effect : a Contribution to Empirical Multifractal Analysis

Abstract : Multifractal analysis is becoming a standard tool in signal processing commonly involved in classical tasks such as detection, estimation or identification. Essentially, in practice, it amounts to measuring a collection of scaling law exponents. It has generally been thought by practitioners that these scaling exponents were related to the details of the multiplicative construction underlying the definitions of most known and used multifractal processes. However, recent results show that these scaling exponents necessarily behave as a linear function of the statistical orders q, for large q s. This confusing association has often been misleading in the use of scaling exponents for real-life data analysis. The present work contributes to the analysis and understanding of this linearization effect and hence to a clarification of this improper association. It is shown that this effect can be explained through an argument involving extreme values and the intrinsic heavy tail nature of the marginal distributions and dependence structure of multifractal processes. These issues are analyzed by means of numerical simulations conducted over specific multifractal processes, the compound Poisson motions (CPM).