Convergence of the structure function of a Multifractal Random Walk in a mixed asymptotic setting

TL;DR

This paper investigates the asymptotic behavior of the structure function of a multifractal random walk, establishing convergence properties in a mixed asymptotic setting where both observation length and sampling frequency grow.

Contribution

It extends previous results on multifractal processes by analyzing convergence of the structure function in a more general mixed asymptotic framework.

Findings

Almost sure convergence of the structure function

L^1 convergence of the structure function

Results analogous to Mandelbrot cascades in a broader setting

Abstract

Some asymptotic properties of a Brownian motion in multifractal time, also called multifractal random walk, are established. We show the almost sure and $L^1$ convergence of its structure function. This is an issue directly connected to the scale invariance and multifractal property of the sample paths. We place ourselves in a mixed asymptotic setting where both the observation length and the sampling frequency may go together to infinity at different rates. The results we obtain are similar to the ones that were given by Ossiander and Waymire and Bacry \emph{et al.} in the simpler framework of Mandelbrot cascades.