*(English)*Zbl 1164.60009

A coupon collector samples with replacement $n$ distinct coupons (each with equal probability $1/n$). Let ${m}_{n}\in \{0,\cdots ,n-1\}$. The sampling is repeated until $n-{m}_{n}$ distinct coupons are collected for the first time. The waiting time, i.e., the random number of draws until ${m}_{n}$ coupons are obtained is denoted by ${W}_{n}\left({m}_{n}\right)$. Let ${\mu}_{n}\left({m}_{n}\right)$ and ${\sigma}_{n}^{2}\left({m}_{n}\right)$ denote mean and variance of ${W}_{n}\left({m}_{n}\right)$ and let ${F}_{n,{m}_{n}}$ denote the distribution function of the normalized random variable $({W}_{n}\left({m}_{n}\right)-{\mu}_{n}\left({m}_{n}\right))/{\sigma}_{n}\left({m}_{n}\right)$.

First limit theorems (for $n\to \infty )$ had been proved by *P. Erdős* and *A. Rényi* [Publ. Math. Inst. Hung. Acad. Sci., Ser. A 6, 215–220 (1961; Zbl 0102.35201)] (for ${m}_{n}=0$) and by *L. E. Baum* and *P. Billingsley* [Ann. Math. Stat. 36 1835–1839 (1965; Zbl 0227.62010)] (for ${m}_{n}=n$). In the second paper, it was also shown that $\left\{{F}_{n\xb7{m}_{n}}\right\}$ is asymptotically normal.

In the paper under review, the author obtains rates of convergence for the asymptotic if ${m}_{n}$ and $\left(n-{m}_{n}\right)/\sqrt{n}$ tend to $\infty $.

##### MSC:

60F05 | Central limit and other weak theorems |