Asymptotic regimes for the occupancy scheme of multiplicative cascades

Abstract

In the classical occupancy scheme, one considers a fixed discrete probability measure $\mathbf{p}=(p_i:i\in \mathcal{I})$ and throws balls independently at random in boxes labeled by $\mathcal{I}$, 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 $(\mathbf{p}(k):k\in \mathbb{N})$ of random probability measures which arise from some multiplicative cascade. Our motivation comes from the study of the asymptotic behavior of certain fragmentation chains