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**1 - 2**of**2**### Theoretical Statistics and Mathematics Unit

, 2013

"... In this work we introduce a new urn model with infinite but countably many colors indexed by an appropriate infinite set. We mainly focus on d-dimensional integer lattice and replacement matrix associated with bounded increment random walks on it. We prove central and local limit theorems for the ex ..."

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In this work we introduce a new urn model with infinite but countably many colors indexed by an appropriate infinite set. We mainly focus on d-dimensional integer lattice and replacement matrix associated with bounded increment random walks on it. We prove central and local limit theorems for the expected configuration of the urn and show that irrespective of the null recurrent or transient behavior of the underlying random walk, the urn models have universal scaling and centering giving appropriate normal distribution at the limit. The work also provides similar results for urn models corresponding to other infinite lattices.

### Pólya urn schemes with infinitely many colors

, 2013

"... Abstract. In this work we introduce a new type of urn model with infinite but countable many colors indexed by an appropriate infinite set. We mainly consider the indexing set of colors to be the d-dimensional integer lattice and consider balanced replacement schemes associated with bounded incremen ..."

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Abstract. In this work we introduce a new type of urn model with infinite but countable many colors indexed by an appropriate infinite set. We mainly consider the indexing set of colors to be the d-dimensional integer lattice and consider balanced replacement schemes associated with bounded increment random walks on it. We prove central and local limit theorems for the random color of the n-th selected ball and show that irrespective of the null recurrent or transient behavior of the underlying random walks, the asymptotic distribution is Gaussian after appropriate centering and scaling. We show that the order of any non-zero centering is always O (logn) and the scaling is O (√logn). The work also provides similar results for urn models with infinitely many colors indexed by more general lattices in Rd. We introduce a novel technique of representing the random color of the n-th selected ball as a suitably sampled point on the path of the underlying random walk.