Limit processes in a model of unequal probabilities arrangement of particles in cells

Let $n_1+n_2+\dots+n_t$ particles be arranged at random into $N$ cells, each of $n_m$ particles getting into the $k$-th cell with a probability $a_k^{(m)}$ ($k=1,2,\dots,N$; $m=1,2,\dots,t$). Let $\mu_0(n)$ be the number of empty cells after $n$ particles have been arranged. We regard $\mu_0(n)$ as a random function of the time parameter $n$, convergence of $\mu_0(n)$ to some– Gaussian or Poisson processes as $n$, $N\to\infty$ being proved.