2 5 Se p 20 08 Small counts in the infinite occupancy scheme

The paper is concerned with the classical occupancy scheme with infinitely many boxes, in which n balls are thrown independently into boxes 1, 2,. . ., with probability pj of hitting the box j, where p1 ≥ p2 ≥. .. > 0 and P ∞ j=1 pj = 1. We establish joint normal approximation as n → ∞ for the numbers of boxes containing r1, r2,. .. , rm 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 r-counts only holds under a condition of regular variation, thus giving a complete characterization of possible limit correlation structures.