Limit Theorems for Some Random Variables Associated with Urn Models

Abstract

Balls are successively thrown, independently and uniformly, in $n$ given urns. Let $N_{n,m}$ be the number of throws required to obtain at least $m$ balls in each urn. Let $N'_{n,m,r}$ be the number of urns containing exactly $r$ balls upon completion of the $N_{n,m}$th throw, $r \geq m$. We prove that, given $N_{n,m} = \lbrack n \log n + (m - 1)n \log \log n + nx\rbrack, N'_{n,m,r} \sim e^{-x} (\log n)^{r-m+1}/r!$ as $n \rightarrow \infty$ in probability. From this, we derive the following limit law for the joint distribution of $N_{n,1}, \cdots, N_{n,m}: \lim_{n \rightarrow\infty} P(N_{n,i} \leq n \log n + (i - 1)n \log \log n + nx_i; 1 \leq i \leq m) = \prod^m_{i=1} \exp(-(1/(i - 1)!)e^{-x_i})$. This result generalizes earlier work of Erdos and Renyi who obtained the limit law for $N_{n,m}$ as $n \rightarrow \infty$.