A randomly reinforced urn

We study an urn containing balls of two or more colors. The urn is sequentially sampled. Each time a ball is extracted from the urn it is reintroduced in it together with a random number of balls of the same color: the distribution of the number of added balls may depend on the color extracted. We prove asymptotic results for the process of colors generated by the urn and for the process of its compositions. Applications to sequential clinical trials are considered as well as connections with response-adaptive design of experiments in a Bayesian framework.