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