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.

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.

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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: | 05 Apr 2016 13:22 |

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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