Small counts in the infinite occupancy scheme

TL;DR

This paper investigates the distribution of small counts in an infinite occupancy scheme, establishing joint normal approximation for counts with minimal variance assumptions and characterizing conditions for convergence.

Contribution

It provides the first joint normal approximation for small counts in the infinite occupancy scheme under broad variance conditions and characterizes when convergence occurs.

Findings

Joint normal approximation holds when variances tend to infinity.

Convergence of counts depends on regular variation conditions.

Complete characterization of limit correlation structures.

Abstract

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 $p_j$ of hitting the box $j$, where $p_1\geq p_2\geq...>0$ and $\sum_{j=1}^\infty p_j=1$. We establish joint normal approximation as $n\to\infty$ for the numbers of boxes containing $r_1,r_2,...,r_m$ 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.