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The Generalised Coupon Collector Problem

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Peter Neal
Peter Neal*
Affiliation:
University of Manchester
*
∗Postal address: School of Mathematics, University of Manchester, Alan Turing Building, Oxford Road, Manchester M13 9PL, UK. Email address: p.neal-2@manchester.ac.uk
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Abstract

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Coupons are collected one at a time from a population containing n distinct types of coupon. The process is repeated until all n coupons have been collected and the total number of draws, Y, from the population is recorded. It is assumed that the draws from the population are independent and identically distributed (draws with replacement) according to a probability distribution X with the probability that a type-i coupon is drawn being P(X = i). The special case where each type of coupon is equally likely to be drawn from the population is the classic coupon collector problem. We consider the asymptotic distribution Y (appropriately normalized) as the number of coupons n → ∞ under general assumptions upon the asymptotic distribution of X. The results are proved by studying the total number of coupons, W(t), not collected in t draws from the population and noting that P(Y ≤ t) = P(W(t) = 0). Two normalizations of Y are considered, the choice of normalization depending upon whether or not a suitable Poisson limit exists for W(t). Finally, extensions to the K-coupon collector problem and the birthday problem are given.

Keywords

Coupon collector problemPoisson convergencebirthday problem

MSC classification

Secondary: 60F05: Central limit and other weak theorems 60G70: Extreme value theory; extremal processes
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 3 , September 2008 , pp. 621 - 629
Copyright
Copyright © Applied Probability Trust 2008 

References

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