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Limiting Distributions for a Class Of Diminishing Urn Models

Part of: Combinatorial probability

Published online by Cambridge University Press:  04 January 2016

Markus Kuba  and
Alois Panholzer
Markus Kuba*
Affiliation:
Technische Universität Wien
Alois Panholzer*
Affiliation:
Technische Universität Wien
*
∗ Postal address: Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien Wiedner Hauptstr. 8-10/104, 1040 Wien, Austria.
∗ Postal address: Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien Wiedner Hauptstr. 8-10/104, 1040 Wien, Austria.
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Abstract

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In this work we analyze a class of 2 × 2 Pólya-Eggenberger urn models with ball replacement matrix and c = pa with . 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 and c = 0. Moreover, we obtain a recursive description of the moment sequence for a generalized problem.

Keywords

Pólya-Eggenberger urn modelpills problemsampling without replacementdiminishing urn

MSC classification

Primary: 60C05: Combinatorial probability
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 44 , Issue 1 , March 2012 , pp. 87 - 116
Copyright
© Applied Probability Trust 

Footnotes

This work was supported by the Austrian Science Foundation FWF, grant number S9608-N13.

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