Asymptotic regimes for the occupancy scheme of multiplicative cascades

TL;DR

This paper investigates the long-term behavior of an occupancy scheme influenced by multiplicative cascades, revealing how random probability measures evolve asymptotically in complex fragmentation processes.

Contribution

It introduces a new asymptotic analysis framework for occupancy schemes driven by multiplicative cascades, extending classical models to more complex, random, and refining regimes.

Findings

Characterization of asymptotic regimes for multiplicative cascade-induced occupancy schemes

Insights into the behavior of fragmentation chains in probabilistic models

Extension of classical occupancy results to random, evolving probability measures

Abstract

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