Three Limit Theorems for Scores Based on Occupancy Numbers

Abstract

Let $N$ balls be distributed independently and at random into $n$ boxes. Let $\rho_{nj}$ denote the number of balls in the $j$th box. Let $(c_0, c_1, c_2, \cdots)$ be a sequence of real numbers. Three limit theorems are proved for the sum $\sum^n_{j=1}c_{\rho_{nj}}$ as $N$ and $n$ tend to infinity in such a way that $N/n \rightarrow 0$.