## Advances in Applied Probability

- Adv. in Appl. Probab.
- Volume 44, Number 2 (2012), 429-451.

### On the joint behavior of types of coupons in generalized coupon collection

Hosam M. Mahmoud and Robert T. Smythe

#### Abstract

The 'coupon collection problem' refers to a class of occupancy problems in
which *j* identical items are distributed, independently and at random, to
*n* cells, with no restrictions on multiple occupancy. Identifying the
cells as coupons, a coupon is 'collected' if the cell is occupied by one or
more of the distributed items; thus, some coupons may never be collected,
whereas others may be collected once or twice or more. We call the number of
coupons collected exactly *r* times *coupons of type r*. The coupon
collection model we consider is general, in that a random number of purchases
occurs at each stage of collecting a large number of coupons; the sample sizes
at each stage are independent and identically distributed according to a
*sampling distribution*. The joint behavior of the various types is an
intricate problem. In fact, there is a variety of joint central limit theorems
(and other limit laws) that arise according to the interrelation between the
mean, variance, and range of the sampling distribution, and of course the phase
(how far we are in the collection processes). According to an appropriate
combination of the mean of the sampling distribution and the number of
available coupons, the phase is sublinear, linear, or superlinear. In the
sublinear phase, the normalization that produces a Gaussian limit law for
uncollected coupons can be used to obtain a multivariate central limit law for
at most two other types - depending on the rates of growth of the mean and
variance of the sampling distribution, we may have a joint central limit
theorem between types 0 and 1, or between types 0, 1, and 2. In the linear
phase we have a multivariate central limit theorem among the types
0, 1,..., *k* for any fixed *k*.

#### Article information

**Source**

Adv. in Appl. Probab., Volume 44, Number 2 (2012), 429-451.

**Dates**

First available in Project Euclid: 16 June 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.aap/1339878719

**Digital Object Identifier**

doi:10.1239/aap/1339878719

**Mathematical Reviews number (MathSciNet)**

MR2977403

**Zentralblatt MATH identifier**

1262.60011

**Subjects**

Primary: 60C05: Combinatorial probability 60F05: Central limit and other weak theorems 05A05: Permutations, words, matrices

Secondary: 60G42: Martingales with discrete parameter 60G48: Generalizations of martingales 60E05: Distributions: general theory

**Keywords**

Urn model random structure stochastic process multivariate martingale multivariate central limit theorem sampling coupon collection phase phase transition

#### Citation

Mahmoud, Hosam M.; Smythe, Robert T. On the joint behavior of types of coupons in generalized coupon collection. Adv. in Appl. Probab. 44 (2012), no. 2, 429--451. doi:10.1239/aap/1339878719. https://projecteuclid.org/euclid.aap/1339878719