Predictive constructions based on measure-valued Pólya urn processes

Measure-valued Pólya urn processes (MVPP) are Markov chains with an additive structure that serve as an extension of the generalized k-color Pólya urn model towards a continuum of possible colors. We prove that, for any MVPP (µn)n≥0 on a Polish space X, the normalized sequence (µn/µn(X))n≥0 agrees with the marginal predictive distributions of some random process (Xn)n≥1. We study the asymptotic properties of the predictive distributions and the empirical frequencies of (Xn)n≥1 under different assumptions. We also investigate a generalization of the above models via a randomization of the law of the reinforcement.