Limit theorems in an occupancy problem

G. I. Ivčenko

Teoriâ veroâtnostej i ee primeneniâ, Tome 16 (1971) no. 2, pp. 292-305

Résumé

There are $N$ cells into which particles are thrown independently of each other. Each particle falls into any fixed cell with probability $1/N$. Let $\nu_m(N,k)$ be the number of throwings after which $k$ cells will contain for the first time at least $m$ particles each. This paper deals with the study of asymptotic behaviour of $\nu_m(N,k)$ as $N\to\infty$ under different assumptions about parameters $k$ and $m$. Limit distributions of $\nu_m(N,k)$ ($N\to\infty$) are found in the case when $m\to\infty$, $m/ln N\le C<\infty$, and either $k=\operatorname{const}$ or $N-k=\operatorname{const}$, or $k/N=\operatorname{const}$.