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 extension of the generalized k-color model towards a continuum possible colors. We prove that, for any MVPP (μn)n≥0 on Polish space X, normalized sequence (μn/μn(X))n≥0 agrees marginal predictive distributions some random process (Xn)n≥1. Moreover, μn=μn−1+RXn, n≥1, where x↦Rx is transition kernel X; thus, if μn−1 represents contents urn, then Xn denotes color ball drawn distribution μn−1/μn−1(X) and RXn—the subsequent reinforcement. In case RXn=WnδXn, non-negative weights W1,W2,…, (Xn)n≥1 better understood randomly reinforced Blackwell MacQueen’s sequence. study asymptotic properties empirical frequencies under different assumptions weights. also investigate generalization above models via randomization law