Strong Convergence of Infinite Color Balanced Urns Under Uniform Ergodicity

Abstract

We consider the generalization of the Pólya urn scheme with possibly infinite many colors as introduced in \cite{Th-Thesis, BaTH2014, BaTh2016, BaTh2017}. For countable many colors, we prove almost sure convergence of the urn configuration under \emph{uniform ergodicity} assumption on the associated Markov chain. The proof uses a stochastic coupling of the sequence of chosen colors with a \emph{branching Markov chain} on a weighted \emph{random recursive tree} as described in \cite{BaTh2017, Sv_2018}. Using this coupling we estimate the covariance between any two selected colors. In particular, we reprove the limit theorem for the classical urn models with finitely many colors.

13 pages

References in corpus (4)

• A survey of random processes with reinforcement
• Asymptotics in randomized urn models
• Multicolor urn models with reducible replacement matrices
• The dominating colour of an infinite Pólya urn model