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Two poisson limit theorems for the coupon collector’s problem with group drawings

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  22 November 2021

Judith Schilling  and
Norbert Henze
Judith Schilling*
Affiliation:
Technische Universität Darmstadt
Norbert Henze*
Affiliation:
Karlsruhe Institute of Technology
*
*Postal address: Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. 7, D-64289 Darmstadt, Germany.
**Postal address: Institute of Stochastics, Karlsruhe Institute of Technology (KIT), Englerstr. 2, D-76133 Karlsruhe, Germany.
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Abstract

In the collector’s problem with group drawings, s out of n different types of coupon are sampled with replacement. In the uniform case, each s-subset of the types has the same probability of being sampled. For this case, we derive a Poisson limit theorem for the number of types that are sampled at most $c-1$ times, where $c \ge 1$ is fixed. In a specified approximate nonuniform setting, we prove a Poisson limit theorem for the special case $c=1$ . As corollaries, we obtain limit distributions for the waiting time for c complete series of types in the uniform case and a single complete series in the approximate nonuniform case.

Keywords

Coupon collector’s problemgroup drawingsPoisson limit theorem

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60G70: Extreme value theory; extremal processes
Type
Original Article
Information
Journal of Applied Probability , Volume 58 , Issue 4 , December 2021 , pp. 966 - 977
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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