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1 results

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

  • Part of
  • Aristides V. Doumas, Vassilis G. Papanicolaou
  • Journal:
    Advances in Applied Probability / Volume 44 / Issue 1 / March 2012
    Published online by Cambridge University Press:
    04 January 2016, pp. 166-195
    Print publication:
    March 2012
    • Article
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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.