On nested infinite occupancy scheme in random environment

TL;DR

This paper studies a complex nested occupancy scheme with random probabilities, deriving a multivariate limit theorem that extends classical results for regenerative partitions, providing new insights into hierarchical fragmentation processes.

Contribution

It introduces a multivariate functional limit theorem for nested occupancy counts, generalizing existing results for Ewens' and regenerative partitions in a hierarchical random environment.

Findings

Established a multivariate functional limit theorem for occupancy counts.

Generalized the functional central limit theorem for regenerative partitions.

Extended classical results to nested hierarchical fragmentation models.

Abstract

We consider an infinite balls-in-boxes occupancy scheme with boxes organised in nested hierarchy, and random probabilities of boxes defined in terms of iterated fragmentation of a unit mass. We obtain a multivariate functional limit theorem for the cumulative occupancy counts as the number of balls approaches infinity. In the case of fragmentation driven by a homogeneous residual allocation model our result generalises the functional central limit theorem for the block counts in Ewens' and more general regenerative partitions.