Advances in Applied Probability
- Adv. in Appl. Probab.
- Volume 44, Number 1 (2012), 166-195.
The coupon collector's problem revisited: asymptotics of the variance
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.
Adv. in Appl. Probab., Volume 44, Number 1 (2012), 166-195.
First available in Project Euclid: 8 March 2012
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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