Asymptotic Normality in a Classical Problem with Balls

B. A. Sevast'yanov   ; V. P. Čistyakov

Teoriâ veroâtnostej i ee primeneniâ, Tome 9 (1964) no. 2, pp. 223-237

Résumé

Each of $n$ balls is deposited in a cell selected at random out of $N$ given cells.The probability of one cell being selected is equal to ${1/N}$, with the successive selections being mutually independent. Let $0\leqq r_1<r_2<\dots<r_s$ be arbitrary fixed integers. The symbol $\mu _r$ denotes a random variable representing the number of those cells that contain exactly $r$ balls. In [3] I. Weiss has proved the integral normal theorem for $\mu_0$ by the method of moments. In this paper we prove the local normal theorem for the random vector $(\mu _{r_1},\dots,\mu _{r_s})$ when $N$, $n\to\infty$ and $0<\alpha _0\leqslant n/{N \leqq\alpha_1}<\infty$ ($\alpha_0$, $\alpha_1$ are constants). In the proof we use the saddle-point method.