On the large-deviation asymptotics in an allocation scheme of particles into distinguishable cells with restrictions on the size of the cells

Equiprobable allocation schemes of allocation of $n$ distinguishable or indistinguishable particles into $N$ distinguishable cells are considered under the condition that the number of particles contained in any cell does not exceed a constant $p\in\mathbf{N}$. Local and integral large-deviation theorems are obtained which estimate the tails of the distributions of the random variable equal to the number of empty cells. The asymptotic behavior of the expectation and variance of the random variable are investigated and a local normal limit theorem is proved for the probabilities of this random variable in the central domain of changing the parameters $n,N$, when $n,N\to\infty$ in such a way that $0 < \alpha_1\le\alpha=n/N\le\alpha_2 < p$ $(\alpha_1,\alpha_2$ are constants).