A Case of Uniform Local Limit Theorems with Changing Lattice in a Classical Problem with Balls

Each of $n$ balls is deposited in a cell selected at random out of $N$ given cells. The successive selections are mutually independent and the probability of any fixed cell to be selected is equal to $1/N$. Let $\mu_r$ be the number of cells that contain exactly $r$ balls, $r=0,1,\dots,n$. In [1] the speed of convergence of the distributions of $\mu_r$ to limiting distributions as $n$, $N\to\infty$ was studied. It was found that the distributions of $\mu_r$ have similar behaviour and converge for any fixed $r$ to either normal or Poisson distribution. The only exception was the behaviour of $\mu_1$ in the case $n$, $N\to\infty$ and $n/N\to0$. In this paper we consider that exceptional case. The main feature is the transition of the distribution of $n-\mu_1$ from the lattice of all non-negative integers to the lattice of even non-negative integers as ratio $n^2/N^3$ is varying from $\infty$ to 0.