Strong convergence for urn models with reducible replacement policy

Romain Abraham (1) , Jean-Stephane Dhersin (2) , Bernard Ycart (3)

1 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
2 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5
3 SMS - Statistique et Modélisation Stochatisque

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.

Mots clés: Urn model; Markov chain; strong convergence

Domaines: Probabilités [math.PR]

Romain Abraham, Jean-Stephane Dhersin, Bernard Ycart. Strong convergence for urn models with reducible replacement policy. Journal of Applied Probability, 2007, 44 (3), pp.652-660. ⟨10.1239/jap/1189717535⟩. ⟨hal-00087017⟩