Refined scaling hypothesis for anomalously diffusing processes

Anomalous diffusion in artificial and natural stochastic processes is studied through the statistics of small-scale fluctuations. It is shown that the moments of certain locally averaged quantities, such as the square or absolute increments, do not scale like power laws, as generally assumed. A much improved scaling function is deduced, in analogy with a procedure first applied to nearest-neighbour dimension estimators. Extremely accurate determination of the scaling exponents is thus possible. Our refined formula is immediately applicable to the analysis of time series in turbulence, physiology, or economics.