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Strong Convergence for URN Models with Reducible Replacement Policy

Part of: Limit theorems

Published online by Cambridge University Press:  14 July 2016

R. Abraham ,
J. S. Dhersin  and
B. Ycart
R. Abraham*
Affiliation:
Université d'Orléans
J. S. Dhersin*
Affiliation:
Université René Descartes
B. Ycart*
Affiliation:
Université Joseph Fourier
*
∗Postal address: MAPMO, CNRS UMR 6628, Université d'Orléans, France. Email address: romain.abraham@univ-orleans.fr
∗∗Postal address: MAP5, CNRS UMR 8145, Université René Descartes, Paris, France. Email address: jean-stephane.dhersin@univ-paris5.fr
∗∗∗Postal address: LJK, CNRS UMR 5224, Université Joseph Fourier, Grenoble, France. Email address: bernard.ycart@ujf-grenoble.fr
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Abstract

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A multitype urn scheme with random replacements is considered. Each time a ball is picked, another ball is added, and its type is chosen according to the transition probabilities of a reducible Markov chain. The vector of frequencies is shown to converge almost surely to a random element of the set of stationary measures of the Markov chain. Its probability distribution is characterized as the solution to a fixed point problem. It is proved to be Dirichlet in the particular case of a single transient state to which no return is possible. This is no longer the case, however, as soon as returns to transient states are allowed.

Keywords

Urn modelMarkov chainstrong convergence

MSC classification

Secondary: 60F15: Strong theorems
Type
Research Article
Information
Journal of Applied Probability , Volume 44 , Issue 3 , September 2007 , pp. 652 - 660
Copyright
Copyright © Applied Probability Trust 2007 

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