A Gaussian process approximation for two-color randomly reinforced urns

Abstract

The Polya 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. The randomly reinforced urn was recently proposed. In this paper, we prove a Gaussian process approximation for the sequence of random composotions of a two-color randomly reinforced urn for both the cases with the equal and unequal reinforcement means. The Gaussian process is a tail stochastic integral with respect to a Brownian motion. 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 prove that the limit distribution of the normalized urn composition has no points masses both when the reinforcements means are equal and unequal under the assumption of only finite $(2+\epsilon)$-th moments.