A new approach to Pólya urn schemes and its infinite color generalization

Abstract

In this work, we introduce a generalization of the classical Pólya urn scheme (Ann. Inst. Henri Poincaré 1 (1930) 117–161) with colors indexed by a Polish space, say, S. The urns are defined as finite measures on S endowed with the Borel σ-algebra, say, $\mathcal{S}$. The generalization is an extension of a model introduced earlier by Blackwell and MacQueen (Ann. Statist. 1 (1973) 353–355). We present a novel approach of representing the observed sequence of colors from such a scheme in terms an associated branching Markov chain on the random recursive tree. The work presents fairly general asymptotic results for this new generalized urn models. As special cases, we show that the results on classical urns, as well as, some of the results proved recently for infinite color urn models in (Bernoulli 23 (2017) 3243–3267; Statist. Probab. Lett. 92 (2014) 232–240), can easily be derived using the general asymptotic. We also demonstrate some newer results for infinite color urns.