P\'olya Urn Schemes with Infinitely Many Colors
Antar Bandyopadhyay, Debleena Thacker

TL;DR
This paper introduces a new infinite-color urn model based on lattice-indexed colors, proving Gaussian limit theorems for the color distribution of the nth ball, regardless of the underlying walk's recurrence properties.
Contribution
It develops a novel representation of the urn's color as a sampled point on a random walk path, enabling new limit theorems for infinite-color urns.
Findings
Asymptotic distribution is Gaussian after centering and scaling.
Order of centering is logarithmic in n.
Order of scaling is square root of logarithm in n.
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 -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…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
