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Zbl 1164.60009
Pósfai, Anna
Rates of convergence for normal approximation in incomplete coupon collection. (English)
[J] Acta Sci. Math. 73, No. 1-2, 333-348 (2007). ISSN 0001-6969

A coupon collector samples with replacement $n$ distinct coupons (each with equal probability $1/n$). Let $m_n\in \{0, \dots ,n-1\}$. 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 $W_n(m_n)$. Let $\mu_n(m_n)$ and $\sigma^2_n(m_n)$ denote mean and variance of $W_n(m_n)$ and let $F_{n,m_n}$ denote the distribution function of the normalized random variable $(W_n(m_n) -\mu_n(m_n))/\sigma_n(m_n)$. \par First limit theorems (for $n\to\infty)$ had been proved by {\it P. Erdös} and {\it A. Rényi} [Publ. Math. Inst. Hung. Acad. Sci., Ser. A 6, 215--220 (1961; Zbl 0102.35201)] (for $m_n=0$) and by {\it L. E. Baum} and {\it 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. \par 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 $\infty$.
[Wilfried Hazod (Dortmund)]

MSC 2000:: *60F05 Weak limit theorems

Keywords: normal approximation; sampling with replacement

Citations: Zbl 0227.62010; Zbl 0102.35201

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