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.
Highlights
In the infinite multinomial occupancy scheme balls are thrown independently in a series of boxes, so that each ball hits box k = 1, 2, . . . with probability pk, where pk > 0 and k∈N pk = 1
Relevant to the subject of this paper, are not sensitive to the labelling of boxes but rather only depend on the integer partition of n comprised of nonzero occupancy numbers
We prove a multivariate functional limit theorem (Theorem 2.1) applicable to the fragmentation laws representable by homogeneous residual allocations models and some other models where the sequence of Pk ’s arranged in decreasing order approaches zero sufficiently fast
Summary
In the infinite occupancy scheme in a random environment the (hitting) probabilities of boxes are positive random variables (Pk)k∈N with an arbitrary joint distribution satisfying k∈N Pk = 1 almost surely (a.s.). On (Pk)k∈N, balls are thrown independently, with probability Pk of hitting box k. Instances of this general setup have received considerable attention within the circle of questions around exchangeable partitions, discrete random measures and their applications to population genetics, Bayesian statistics and computer science. We prove a multivariate functional limit theorem (Theorem 2.1) applicable to the fragmentation laws representable by homogeneous residual allocations models (including the GEM/PD distribution) and some other models where the sequence of Pk ’s arranged in decreasing order approaches zero sufficiently fast. A univariate functional limit for (Kn,1(s))s∈[0,1] in the case of Bernoulli sieve was previously obtained in [2]
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