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

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.