A Gaussian process approximation for two-color randomly reinforced urns∗

The Pólya urn has been extensively studied and is widely applied in many disciplines. An important application is to use urn models to develop randomized treatment allocation schemes in clinical studies. As an extension of the Pólya urn, the randomly reinforced urn was recently proposed to optimize clinical trials in the sense that patients are assigned to the best treatment with probability converging to one. In this paper, we prove a Gaussian process approximation for the sequence of random compositions of a two-color randomly reinforced urn for both the cases with the equal and unequal reinforcement means. By using the Gaussian approximation, the law of the iterated logarithm and the functional central limit theorem in both the stable convergence sense and the almost-sure conditional convergence sense are established. Also as a consequence, we are able to to prove that a random limit of the normalized urn composition has no point masses under the only assumption of finite (2 + )-th moments.