Convergence of the variables $\mu_r(n)$ to Gaussian and Poisson processes in the classical problem with balls

Let $n$ balls be distributed at random in $N$ boxes. Each ball may fall into any box with the same probability $1/N$ independently of the others. Let $\mu_r(n)$ be the number of boxes which contain exactly $r$ balls $(r=1,2,\dots)$. We consider $\mu_r(n)$ as a random function of the time parameter $n$. In this paper we prove that the random function $\mu_r(n)$ converges to some Gaussian or Poisson process as $n$, $N\to\infty$.