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.

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.

Citations

5 citations in Web of Science®
7 citations in Scopus®

Detailed statistics

Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics English 9 February 2009 16 Nov 2009 20:25 05 Apr 2016 13:22 Institute of Mathematical Statistics 1083-6489 http://www.emis.de/journals/EJP-ECP/_ejpecp/viewarticle77b2.html?id=1912 http://arxiv.org/abs/0809.4387http://www.math.washington.edu/~ejpecp/
Permanent URL: http://doi.org/10.5167/uzh-21237

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