A NEW APPROACH TO POLYA URN SCHEMES AND ITS INFINITE COLOR GENERALIZATION

• 成果类型: Article
• 署名作者: Bandyopadhyay, Antar; Thacker, Debleena
• 署名单位: Indian Statistical Institute; Indian Statistical Institute Kolkata; Durham University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/21-AAP1671
• 发表日期: 2022
• 页码: 46-79
• 关键词: branching-processes; strong-convergence; LIMIT-THEOREMS; strong laws; random-walk; models; attachment

摘要

In this work, we introduce a generalization of the classical Polya urn scheme (Ann. Inst. Henri Poincare 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 sigma-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.