  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.
