Title: Urn models with reducible replacements
Author: R. Abraham, J.-S. Dhersin, B. Ycart
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 more the case as soon as
returns to transient states are allowed.
Keywords: urn model, Markov chain, strong convergence
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AMS Subject Classification: 60F15