Small counts in the infinite occupancy scheme

2008/09/25 by A. D. Barbour, Alexander Gnedin, Barbour, A. D. +1
Mathematics · #60F05 #60F15 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.0809.4387

openalex publication_date 2008/09/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The paper is concerned with the classical occupancy scheme with infinitely many boxes, in which n balls are thrown independently into boxes 1,2,..., with probability p_j of hitting the box j, where p₁≥ p₂≥...>0 and ∑_j=1^∞ p_j=1. We establish joint normal approximation as n→∞ for the numbers of boxes containing r₁,r₂,...,r_m balls, standardized in the natural way, assuming only that the variances of these counts all tend to infinity. The proof of this approximation is based on a de-Poissonization lemma. We then review sufficient conditions for the variances to tend to infinity. Typically, the normal approximation does not mean convergence. We show that the convergence of the full vector of r-counts only holds under a condition of regular variation, thus giving a complete characterization of possible limit correlation structures.

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