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Approaching the coupon collector’s problem with group drawings via Stein’s method

Part of: Limit theorems Combinatorial probability

Published online by Cambridge University Press:  25 April 2023

Carina Betken  and
Christoph Thäle
Carina Betken*
Affiliation:
Ruhr University Bochum
Christoph Thäle*
Affiliation:
Ruhr University Bochum
*
*Postal address: Faculty of Mathematics, Ruhr University Bochum, 44780 Bochum, Germany.
*Postal address: Faculty of Mathematics, Ruhr University Bochum, 44780 Bochum, Germany.
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Abstract

We study the coupon collector’s problem with group drawings. Assume there are n different coupons. At each time precisely s of the n coupons are drawn, where all choices are supposed to have equal probability. The focus lies on the fluctuations, as $n\to\infty$, of the number $Z_{n,s}(k_n)$ of coupons that have not been drawn in the first $k_n$ drawings. Using a size-biased coupling construction together with Stein’s method for normal approximation, a quantitative central limit theorem for $Z_{n,s}(k_n)$ is shown for the case that $k_n=({n/s})(\alpha\log(n)+x)$, where $0<\alpha<1$ and $x\in\mathbb{R}$. The same coupling construction is used to retrieve a quantitative Poisson limit theorem in the boundary case $\alpha=1$, again using Stein’s method.

Keywords

Central limit theoremcoupon collector’s problemPoisson limit theoremsize-biased couplingStein’s method

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 60F05: Central limit and other weak theorems
Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 4 , December 2023 , pp. 1352 - 1366
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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