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Gaussian phases in generalized coupon collection

Part of: Limit theorems Stochastic processes Combinatorial probability Combinatorics

Published online by Cambridge University Press:  01 July 2016

Hosam M. Mahmoud
Hosam M. Mahmoud*
Affiliation:
The George Washington University
*
∗ Postal address: Department of Statistics, The George Washington University, Washington, DC 20052, USA. Email address: [email protected]
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Abstract

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In this paper we consider a generalized coupon collection problem in which a customer repeatedly buys a random number of distinct coupons in order to gather a large number n of available coupons. We address the following question: How many different coupons are collected after k = kn draws, as n → ∞? We identify three phases of kn: the sublinear, the linear, and the superlinear. In the growing sublinear phase we see o(n) different coupons, and, with true randomness in the number of purchases, under the appropriate centering and scaling, a Gaussian distribution is obtained across the entire phase. However, if the number of purchases is fixed, a degeneracy arises and normality holds only at the higher end of this phase. If the number of purchases have a fixed range, the small number of different coupons collected in the sublinear phase is upgraded to a number in need of centering and scaling to become normally distributed in the linear phase with a different normal distribution of the type that appears in the usual central limit theorems. The Gaussian results are obtained via martingale theory. We say a few words in passing about the high probability of collecting nearly all the coupons in the superlinear phase. It is our aim to present the results in a way that explores the critical transition at the ‘seam line’ between different Gaussian phases, and between these phases and other nonnormal phases.

Keywords

Urn modelrandom structuremartingalecentral limit theoremcoupon collection

MSC classification

Primary: 60C05: Combinatorial probability 60F05: Central limit and other weak theorems 05A05: Permutations, words, matrices
Secondary: 60G42: Martingales with discrete parameter
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 42 , Issue 4 , December 2010 , pp. 994 - 1012
Copyright
Copyright © Applied Probability Trust 2010 

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