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Strong Convergence for URN Models with Reducible Replacement Policy

  • R. Abraham (a1), J. S. Dhersin (a2) and B. Ycart (a3)
Abstract

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.

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Copyright
Corresponding author
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
References
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  • ISSN: 0021-9002
  • EISSN: 1475-6072
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