  In the classical occupancy scheme, one considers a fixed discrete probability
measure ${\bf p}=(p_i: {i\in{\cal I}})$ and throws balls independently at
random in boxes labeled by ${\cal I}$, such that $p_i$ is the probability that
a given ball falls into the box $i$. In this work, we are interested in
asymptotic regimes of this scheme in the situation induced by a refining
sequence $({\bf p}(k) : k\in\N)$ of random probability measures which arise
from some multiplicative cascade. Our motivation comes from the study of the
asymptotic behavior of certain fragmentation chains
