The speed of convergence to limiting distributions 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 tobe selected is equal to $1/N$. Let $\mu_r$ be the number of cells that contain exactly $r$ balls, $r=0,\dots,n$. In this paper we study the speed of convergence of the distributions of $\mu_r$ to limiting distributions as $n$, $N\to\infty$. We calculate the variational distance between the distributions of $\mu_r$ and the limiting distributions. When the order of magnitude of $\alpha=n/N$ is known we find the nearest distribution to $\mu_r$ in the sense of this distanceand calculate its exact upper bound with respect to $\alpha$, $0<\alpha<\infty$. As a result we may compare the speed of convergence of the distributions of цг with the classical case of approximation of the binomial distribution that has been investigated by Yu. V. Prochorov [1]. The exact upper bound of the distance to the nearest limiting distribution for the binomial distribution with parameters $n$ and $p$, $0\le p\le1$, is $cn^{-1/3}(1+O(n^{-1/3}))$ and for the distributions of $\mu_r$ it is equal to $cn^{-1/3}(\log n+(3r+1)\log\log n)^{2/3}(1+O(\log^{-1}n)$ where с is a known constant.