The convergence to a~Gaussian process of the number of empty cells in the classical problem of distributing particles among cells

Yu. V. Bolotnikov

Matematičeskie zametki, Tome 4 (1968) no. 1, pp. 97-103.

Résumé

We consider a case in which $n$ particles are distributed independently of one another in $N$ cells. We examine the behavior of the number of empty cells, $\mu_0(n)$, as a random function of the parameter $n$ when $n,N\to\infty$. We prove that for suitable variation of the time parameter, $\mu_0(n)$ will converge to a Gaussian process in the following cases: a) $n/N\to\infty$, $n/N-\ln N\to-\infty$; b) $n/N\to0$, $n^2/N\to\infty$.