Asymptotic regimes for the occupancy scheme of multiplicative cascades

In the classical occupancy scheme, one considers a fixed discrete probability measure p = ( p i : i ∈ I ) and throws balls independently at random in boxes labeled by 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 ( 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.