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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 {0,,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 μ n (m n ) and σ n 2 (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 )-μ n (m n ))/σ n (m n ).

First limit theorems (for n) 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 F n·m n is asymptotically normal.

In the paper under review, the author obtains rates of convergence for the asymptotic if m n and n-m n /n tend to .

MSC:
60F05Central limit and other weak theorems