Open Access
2009 Small counts in the infinite occupancy scheme
A. Barbour, A. Gnedin
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Electron. J. Probab. 14: 365-384 (2009). DOI: 10.1214/EJP.v14-608


The paper is concerned with the classical occupancy scheme in which balls are thrown independently into infinitely many boxes, with given probability of hitting each of the boxes. We establish joint normal approximation, as the number of balls goes to infinity, for the numbers of boxes containing any fixed number of 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 counts only holds under a condition of regular variation, thus giving a complete characterization of possible limit correlation structures.


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A. Barbour. A. Gnedin. "Small counts in the infinite occupancy scheme." Electron. J. Probab. 14 365 - 384, 2009.


Accepted: 9 February 2009; Published: 2009
First available in Project Euclid: 1 June 2016

zbMATH: 1189.60048
MathSciNet: MR2480545
Digital Object Identifier: 10.1214/EJP.v14-608

Primary: 60F05
Secondary: 60C05

Keywords: Normal approximation , occupancy problem , poissonization , regular variation

Vol.14 • 2009
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