Rate of convergence and large deviation for the infinite color Pólya urn schemes

In this work we consider the infinite color urn model associated with a bounded increment random walk on Zd. This model was first introduced in  Bandyopadhyay and Thacker (2013). We prove that the rate of convergence of the expected configuration of the urn at time n with appropriate centering and scaling is of the order O((logn)−1/2). Moreover we derive bounds similar to the classical Berry-Esseen bound. Further we show that for the expected configuration a large deviation principle (LDP) holds with a good rate function and speed logn.