Rate of Convergence and Large Deviation for the Infinite Color P\'olya Urn Schemes

In this work we consider the \emph{infinite color urn model} associated with a bounded increment random walk on $\Zbold^d$. This model was first introduced by 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 ${\mathcal O}\left(\frac{1}{\sqrt{\log n}}\right)$. Moreover we derive bounds similar to the classical Berry-Essen bound. Further we show that for the expected configuration a \emph{large deviation principle (LDP)} holds with a good rate function and speed $\log n$.