Asymptotic regimes for the occupancy scheme of multiplicative cascades

In the classical occupancy scheme, one considers a fixed discrete probability measure p=(pi:i∈I) and throws balls independently at random in boxes labeled by II, such that pipi is the probability that a given ball falls into the box ii. In this work, we are interested in asymptotic regimes of this scheme in the situation induced by a refining sequence (p(k):k∈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.