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Rates of convergence for normal approximation in incomplete coupon collection. (English) Zbl 1164.60009

A coupon collector samples with replacement $n$ distinct coupons (each with equal probability $1/n$). Let ${m}_{n}\in \left\{0,\cdots ,n-1\right\}$. 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 $\left({W}_{n}\left({m}_{n}\right)-{\mu }_{n}\left({m}_{n}\right)\right)/{\sigma }_{n}\left({m}_{n}\right)$.

First limit theorems (for $n\to \infty \right)$ 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 $\left\{{F}_{n·{m}_{n}}\right\}$ 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
##### Keywords:
normal approximation; sampling with replacement