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

TL;DR

This paper develops bias-reduced estimators for the structure functions of various multifractal processes, proving their asymptotic normality and achieving faster polynomial convergence rates compared to previous methods.

Contribution

It introduces new estimators with polynomial convergence rates for multifractal measures and proves their central limit theorems, improving upon existing biased estimators.

Findings

Bias-reduced estimators exhibit polynomial convergence rates.

Central limit theorems are established for these estimators.

Improved accuracy over previous logarithmic-rate estimators.

Abstract

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.