Estimating the scaling function of multifractal measures and multifractal random walks using ratios

In this paper, we prove central limit theorems for bias reduced estimators of the structure function of several multifractal processes, namely mutiplicative cascades, multifractal random measures, multifractal random walk and multifractal fractional random walk as defined by Lude\~{n}a [Ann. Appl. Probab. 18 (2008) 1138-1163]. Previous estimators of the structure functions considered in the literature were severely biased with a logarithmic rate of convergence, whereas the estimators considered here have a polynomial rate of convergence.