# Strong Convergence of Infinite Color Balanced Urns Under Uniform Ergodicity

## TL;DR

This paper proves almost sure convergence of infinite-color urn models under uniform ergodicity, using a novel coupling with branching Markov chains on random recursive trees, extending classical finite-color results.

## Contribution

It introduces a new convergence proof for infinite-color urns under uniform ergodicity, employing a stochastic coupling with branching Markov chains on recursive trees.

## Key findings

- Proves almost sure convergence of infinite-color urns.
- Establishes covariance estimates between colors.
- Reproves classical finite-color urn limit theorems.

## Abstract

We consider the generalization of the P\'olya 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.