53 research outputs found
P\'olya Urn Schemes with Infinitely Many Colors
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 -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
-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 and the
scaling is . The work also provides
similar results for urn models with infinitely many colors indexed by more
general lattices in . We introduce a novel technique of
representing the random color of the -th selected ball as a suitably sampled
point on the path of the underlying random walk. This helps us to derive the
central and local limit theorems.Comment: 32 pages and 1 figure. Motivation for the work has been added and few
typing errors have been correcte
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