Small counts in the infinite occupancy scheme
A. D. Barbour, University of Zurich
A. V. Gnedin, Utrecht University
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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