Sequential rank and the Pólya urn

1979 ◽  
Vol 16 (1) ◽  
pp. 213-219 ◽  
Author(s):  
Herbert Robbins ◽  
John Whitehead
Keyword(s):  
Random Variables ◽  
Pólya Urn ◽  
Polya Urn ◽  
Urn Scheme ◽  
Sequential Rank ◽  
Independent Identically Distributed

A sequence of independent, identically distributed random variables is observed. After a sample of m has been collected attention is fixed to a particular observation. In this paper the fluctuations of the rank of this observation, as sampling continues, will be studied.The process can be modelled by a Pólya urn scheme, and new results are obtained which are of interest in both sequential rank and Pólya urn contexts.

Sequential rank and the Pólya urn

1979 ◽  
Vol 16 (01) ◽  
pp. 213-219
Author(s):  
Herbert Robbins ◽  
John Whitehead
Keyword(s):  
Random Variables ◽  
Pólya Urn ◽  
Polya Urn ◽  
Urn Scheme ◽  
Sequential Rank ◽  
Independent Identically Distributed

A sequence of independent, identically distributed random variables is observed. After a sample of m has been collected attention is fixed to a particular observation. In this paper the fluctuations of the rank of this observation, as sampling continues, will be studied. The process can be modelled by a Pólya urn scheme, and new results are obtained which are of interest in both sequential rank and Pólya urn contexts.

scholarly journals Beta-Stacy processes and a generalization of the Pólya-urn scheme

1997 ◽  
Vol 25 (4) ◽  
pp. 1762-1780 ◽  
Author(s):  
Stephen Walker ◽  
Pietro Muliere
Keyword(s):  
Pólya Urn ◽  
Polya Urn ◽  
Urn Scheme

Another Polya Urn Scheme: 10504

1998 ◽  
Vol 105 (2) ◽  
pp. 181
Author(s):  
Richard Hamming ◽  
Roger Pinkham ◽  
Richard Stong
Keyword(s):  
Pólya Urn ◽  
Polya Urn ◽  
Urn Scheme

scholarly journals Strong convergence of infinite color balanced urns under uniform ergodicity

2020 ◽  
Vol 57 (3) ◽  
pp. 853-865
Author(s):  
Antar Bandyopadhyay ◽  
Svante Janson ◽  
Debleena Thacker
Keyword(s):  
Markov Chain ◽  
Limit Theorem ◽  
Strong Convergence ◽  
Almost Sure Convergence ◽  
Urn Models ◽  
Recursive Tree ◽  
Pólya Urn ◽  
Uniform Ergodicity ◽  
Polya Urn ◽  
Urn Scheme

AbstractWe consider the generalization of the Pólya urn scheme with possibly infinitely many colors, as introduced in [37], [4], [5], and [6]. For countably many colors, we prove almost sure convergence of the urn configuration under the uniform ergodicity assumption on the associated Markov chain. The proof uses a stochastic coupling of the sequence of chosen colors with a branching Markov chain on a weighted random recursive tree as described in [6], [31], and [26]. Using this coupling we estimate the covariance between any two selected colors. In particular, we re-prove the limit theorem for the classical urn models with finitely many colors.

On central limit and iterated logarithm supplements to the martingale convergence theorem

1977 ◽  
Vol 14 (4) ◽  
pp. 758-775 ◽  
Author(s):  
C. C. Heyde
Keyword(s):  
Central Limit ◽  
Convergence Theorem ◽  
Mean Square ◽  
Martingale Convergence Theorem ◽  
Iterated Logarithm ◽  
Pólya Urn ◽  
Integrable Martingale ◽  
Polya Urn ◽  
Urn Scheme ◽  
Martingale Convergence

Let {Sn, n ≧ 1} be a zero, mean square integrable martingale for which so that Sn → S∞ a.s., say, by the martingale convergence theorem. The paper is principally concerned with obtaining central limit and iterated logarithm results for Bn(Sn – S∞) where the multipliers Bn ↑ ∞ a.s. An example on the Pólya urn scheme is given to illustrate the results.

scholarly journals Identifying differentially expressed genes using the Polya urn scheme

2017 ◽  
Vol 24 (6) ◽  
pp. 627-640
Author(s):  
Erlandson Ferreira Saraiva ◽  
Adriano Kamimura Suzuki ◽  
Luís Aparecido Milan
Keyword(s):  
Differentially Expressed Genes ◽  
Differentially Expressed ◽  
Pólya Urn ◽  
Polya Urn ◽  
Urn Scheme

On central limit and iterated logarithm supplements to the martingale convergence theorem

1977 ◽  
Vol 14 (04) ◽  
pp. 758-775 ◽  
Author(s):  
C. C. Heyde
Keyword(s):  
Central Limit ◽  
Convergence Theorem ◽  
Mean Square ◽  
Martingale Convergence Theorem ◽  
Iterated Logarithm ◽  
Pólya Urn ◽  
Integrable Martingale ◽  
Polya Urn ◽  
Urn Scheme ◽  
Martingale Convergence

Let {Sn , n ≧ 1} be a zero, mean square integrable martingale for which so that Sn → S ∞ a.s., say, by the martingale convergence theorem. The paper is principally concerned with obtaining central limit and iterated logarithm results for Bn (Sn – S∞ ) where the multipliers Bn ↑ ∞ a.s. An example on the Pólya urn scheme is given to illustrate the results.

A Bayesian subgroup analysis with a zero-enriched Polya Urn scheme

2010 ◽  
Vol 30 (4) ◽  
pp. 312-323 ◽  
Author(s):  
S. Sivaganesan ◽  
Purushottam W. Laud ◽  
Peter Müller
Keyword(s):  
Subgroup Analysis ◽  
Pólya Urn ◽  
Polya Urn ◽  
Urn Scheme
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