  This paper investigates new properties concerning the multifractal structure
of a class of statistically self-similar measures. These measures include the
well-known Mandelbrot multiplicative cascades, sometimes called independent
random cascades. We evaluate the scale at which the multifractal structure of
these measures becomes discernible. The value of this scale is obtained through
what we call the growth speed in H\"older singularity sets of a Borel measure.
This growth speed yields new information on the multifractal behavior of the
rescaled copies involved in the structure of statistically self-similar
measures. Our results are useful to understand the multifractal nature of
various heterogeneous jump processes.
