Asymptotic Properties of Multicolor Randomly Reinforced Pólya Urns

Abstract

The generalized Pólya urn has been extensively studied and is widely applied in many disciplines. An important application of urn models is in the development of randomized treatment allocation schemes in clinical studies. The randomly reinforced urn was recently proposed, but, although the model has some intuitively desirable properties, it lacks theoretical justification. In this paper we obtain important asymptotic properties for multicolor reinforced urn models. We derive results for the rate of convergence of the number of patients assigned to each treatment under a set of minimum required conditions and provide the distributions of the limits. Furthermore, we study the asymptotic behavior for the nonhomogeneous case.