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The Coupon Collector's Problem Revisited: Asymptotics of the Variance

Part of: Limit theorems

Published online by Cambridge University Press:  04 January 2016

Aristides V. Doumas  and
Vassilis G. Papanicolaou
Aristides V. Doumas*
Affiliation:
National Technical University of Athens
Vassilis G. Papanicolaou*
Affiliation:
National Technical University of Athens
*
∗ Postal address: Department of Mathematics, National Technical University of Athens, Zografou Campus, 157 80 Athens, Greece.
∗ Postal address: Department of Mathematics, National Technical University of Athens, Zografou Campus, 157 80 Athens, Greece.
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Abstract

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We develop techniques for computing the asymptotics of the first and second moments of the number TN of coupons that a collector has to buy in order to find all N existing different coupons as N → ∞. The probabilities (occurring frequencies) of the coupons can be quite arbitrary. From these asymptotics we obtain the leading behavior of the variance V[TN] of TN (see Theorems 3.1 and 4.4). Then, we combine our results with the general limit theorems of Neal in order to derive the limit distribution of TN (appropriately normalized), which, for a large class of probabilities, turns out to be the standard Gumbel distribution. We also give various illustrative examples.

Keywords

Coupon collector's problemhigher asymptoticslimit distribution

MSC classification

Primary: 60F05: Central limit and other weak theorems 60F99: None of the above, but in this section
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 44 , Issue 1 , March 2012 , pp. 166 - 195
Copyright
© Applied Probability Trust 

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