Advances in Applied Probability

The coupon collector's problem revisited: asymptotics of the variance

Aristides V. Doumas and Vassilis G. Papanicolaou

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Abstract

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.

Article information

Source
Adv. in Appl. Probab., Volume 44, Number 1 (2012), 166-195.

Dates
First available in Project Euclid: 8 March 2012

Permanent link to this document
https://projecteuclid.org/euclid.aap/1331216649

Digital Object Identifier
doi:10.1239/aap/1331216649

Mathematical Reviews number (MathSciNet)
MR2951551

Zentralblatt MATH identifier
1260.60035

Subjects
Primary: 60F05: Central limit and other weak theorems 60F99: None of the above, but in this section

Keywords
Coupon collector's problem higher asymptotics limit distribution

Citation

Doumas, Aristides V.; Papanicolaou, Vassilis G. The coupon collector's problem revisited: asymptotics of the variance. Adv. in Appl. Probab. 44 (2012), no. 1, 166--195. doi:10.1239/aap/1331216649. https://projecteuclid.org/euclid.aap/1331216649


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