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A new approach to Pólya urn schemes and its infinite color generalization

Bandyopadhyay, Antar; Thacker, Debleena

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Authors

Antar Bandyopadhyay



Abstract

In this work, we introduce a generalization of the classical Pólya urn scheme (Ann. Inst. Henri Poincaré 1 (1930) 117–161) with colors indexed by a Polish space, say, S. The urns are defined as finite measures on S endowed with the Borel σ-algebra, say, S . The generalization is an extension of a model introduced earlier by Blackwell and MacQueen (Ann. Statist. 1 (1973) 353–355). We present a novel approach of representing the observed sequence of colors from such a scheme in terms an associated branching Markov chain on the random recursive tree. The work presents fairly general asymptotic results for this new generalized urn models. As special cases, we show that the results on classical urns, as well as, some of the results proved recently for infinite color urn models in (Bernoulli 23 (2017) 3243–3267; Statist. Probab. Lett. 92 (2014) 232–240), can easily be derived using the general asymptotic. We also demonstrate some newer results for infinite color urns.

Citation

Bandyopadhyay, A., & Thacker, D. (2022). A new approach to Pólya urn schemes and its infinite color generalization. Annals of Applied Probability, 32(1), 46-79. https://doi.org/10.1214/21-aap1671

Journal Article Type Article
Online Publication Date Feb 27, 2022
Publication Date 2022-02
Deposit Date Oct 26, 2022
Publicly Available Date Oct 31, 2022
Journal Annals of Applied Probability
Print ISSN 1050-5164
Publisher Institute of Mathematical Statistics
Peer Reviewed Peer Reviewed
Volume 32
Issue 1
Pages 46-79
DOI https://doi.org/10.1214/21-aap1671
Public URL https://durham-repository.worktribe.com/output/1190441

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