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Small counts in the infinite occupancy scheme


Barbour, A D; Gnedin, A (2009). Small counts in the infinite occupancy scheme. Electronic Journal of Probability, 14(13):365-384.

Abstract

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.

Abstract

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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9 citations in Web of Science®
10 citations in Scopus®
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69 downloads since deposited on 16 Nov 2009
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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:9 February 2009
Deposited On:16 Nov 2009 20:25
Last Modified:06 Dec 2017 20:35
Publisher:Institute of Mathematical Statistics
ISSN:1083-6489
Official URL:http://www.emis.de/journals/EJP-ECP/_ejpecp/viewarticle77b2.html?id=1912
Related URLs:http://arxiv.org/abs/0809.4387
http://www.math.washington.edu/~ejpecp/

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