Infinite-color randomly reinforced urns with dominant colors

We define and prove limit results for a class of dominant P\'olya sequences, which are randomly reinforced urn processes with color-specific random weights unbounded number possible colors. Under fairly mild assumptions on the expected reinforcement, we show that predictive empirical distributions converge almost surely (a.s.) in total variation to same probability measure $\tilde{P}$; moreover, $\tilde{P}(\mathcal{D})=1$ a.s., where $\mathcal{D}$ denotes set colors reinforcement is maximum. In general case, probabilities frequencies any $\delta$-neighborhood a.s. one. That is, although non-dominant continue be regularly observed, their distance converges zero. refine above rates convergence. further hint potential applications sequences randomized clinical trials species sampling, use our central Bayesian inference.