  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.
