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.

中文翻译：

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随机环境下嵌套无限占用方案

我们考虑了一个无限的盒子中球占用方案，其中盒子以嵌套层次结构组织，盒子的随机概率根据单位质量的迭代碎片定义。随着球的数量接近无穷大，我们获得了累积占用计数的多元函数极限定理。在由同构残差分配模型驱动的碎片的情况下，我们的结果概括了 Ewens 和更一般的再生分区中块计数的功能中心极限定理。