Scaling exponents estimation for multiscaling processes

We study the statistical performance of multiresolution (wavelet based) estimators commonly used for the estimation of the scaling exponents, ζ(q), of multifractal processes. So far, such studies have been conducted exclusively using the celebrated Mandelbrot´s cascades. A new class of processes, compound Poisson cascades, with better statistical properties -stationary increments and continuous scale invariance - has recently been proposed in the literature. Making use of this new type of process, we show that the multiresolution estimators are characterised by a generic and systematic feature: beyond a critical order, q, (which is determined analytically), they fail to estimate the ζ(q) and present instead a linear behaviour in q. We study in detail this linearisation effect and show that it does not disappear in the limit of infinite observation duration, n, and that the parameters characterising it do not depend on n. We comment on its major practical consequences and on its having been mostly overlooked in applications.