Advances in Applied Probability

Limiting distributions for a class of diminishing urn models

Markus Kuba and Alois Panholzer

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Abstract

In this work we analyze a class of 2 x 2 Pólya-Eggenberger urn models with ball replacement matrix M = (-a 0 \\ c -d), a, dN and c = pa with pN0. We determine limiting distributions by obtaining a precise recursive description of the moments of the considered random variables, which allows us to deduce asymptotic expansions of the moments. In particular, we obtain limiting distributions for the pills problem a = c = d = 1, originally proposed by Knuth and McCarthy. Furthermore, we also obtain limiting distributions for the well-known sampling without replacement urn, a = d = 1 and c = 0, and generalizations of it to arbitrary a, dN and c = 0. Moreover, we obtain a recursive description of the moment sequence for a generalized problem.

Article information

Source
Adv. in Appl. Probab., Volume 44, Number 1 (2012), 87-116.

Dates
First available in Project Euclid: 8 March 2012

Permanent link to this document
https://projecteuclid.org/euclid.aap/1331216646

Digital Object Identifier
doi:10.1239/aap/1331216646

Mathematical Reviews number (MathSciNet)
MR2951548

Zentralblatt MATH identifier
1300.60023

Subjects
Primary: 60C05: Combinatorial probability

Keywords
Pólya-Eggenberger urn model pills problem sampling without replacement diminishing urn

Citation

Kuba, Markus; Panholzer, Alois. Limiting distributions for a class of diminishing urn models. Adv. in Appl. Probab. 44 (2012), no. 1, 87--116. doi:10.1239/aap/1331216646. https://projecteuclid.org/euclid.aap/1331216646


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