Order statistics and the Pólya urn process

1977 ◽  
Vol 9 (2) ◽  
pp. 205-206
Author(s):  
Herbert Robbins ◽  
John Whitehead
Keyword(s):  
Order Statistics ◽  
Pólya Urn ◽  
Polya Urn

Order statistics and the Pólya urn process

1977 ◽  
Vol 9 (02) ◽  
pp. 205-206
Author(s):  
Herbert Robbins ◽  
John Whitehead
Keyword(s):  
Order Statistics ◽  
Pólya Urn ◽  
Polya Urn

scholarly journals Predictive Constructions Based on Measure-Valued Pólya Urn Processes

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2845
Author(s):  
Sandra Fortini ◽  
Sonia Petrone ◽  
Hristo Sariev
Keyword(s):  
Polish Space ◽  
Asymptotic Properties ◽  
Transition Kernel ◽  
Urn Model ◽  
Additive Structure ◽  
Predictive Distributions ◽  
Pólya Urn ◽  
Random Weights ◽  
Random Transition ◽  
Polya Urn

Measure-valued Pólya urn processes (MVPP) are Markov chains with an additive structure that serve as an extension of the generalized k-color Pólya urn model towards a continuum of possible colors. We prove that, for any MVPP (μn)n≥0 on a Polish space X, the normalized sequence (μn/μn(X))n≥0 agrees with the marginal predictive distributions of some random process (Xn)n≥1. Moreover, μn=μn−1+RXn, n≥1, where x↦Rx is a random transition kernel on X; thus, if μn−1 represents the contents of an urn, then Xn denotes the color of the ball drawn with distribution μn−1/μn−1(X) and RXn—the subsequent reinforcement. In the case RXn=WnδXn, for some non-negative random weights W1,W2,…, the process (Xn)n≥1 is better understood as a randomly reinforced extension of Blackwell and MacQueen’s Pólya sequence. We study the asymptotic properties of the predictive distributions and the empirical frequencies of (Xn)n≥1 under different assumptions on the weights. We also investigate a generalization of the above models via a randomization of the law of the reinforcement.

Strong convergence of proportions in a multicolor Pólya urn

1997 ◽  
Vol 34 (2) ◽  
pp. 426-435 ◽  
Author(s):  
Raúl Gouet
Keyword(s):  
Strong Convergence ◽  
Convex Combination ◽  
Urn Model ◽  
Pólya Urn ◽  
Polya Urn ◽  
Pólya Urn Model ◽  
Generalized Pólya Urn

We prove strong convergence of the proportions Un/Tn of balls in a multitype generalized Pólya urn model, using martingale arguments. The limit is characterized as a convex combination of left dominant eigenvectors of the replacement matrix R, with random Dirichlet coefficients.

scholarly journals Limit Theorems for Random Triangular URN Schemes

2009 ◽  
Vol 46 (03) ◽  
pp. 827-843 ◽  
Author(s):  
Rafik Aguech
Keyword(s):  
Limit Theorems ◽  
Embedding Method ◽  
Largest Eigenvalue ◽  
Dominant Class ◽  
Pólya Urn ◽  
Polya Urn ◽  
The Mean ◽  
Generalized Pólya Urn ◽  
The Largest Eigenvalue

In this paper we study a generalized Pólya urn with balls of two colors and a random triangular replacement matrix. We extend some results of Janson (2004), (2005) to the case where the largest eigenvalue of the mean of the replacement matrix is not in the dominant class. Using some useful martingales and the embedding method introduced in Athreya and Karlin (1968), we describe the asymptotic composition of the urn after the nth draw, for large n.

Review of pólya Urn models by Hosam Mahmoud

2011 ◽  
Vol 42 (2) ◽  
pp. 27-33
Author(s):  
Stephen Stanhope
Keyword(s):  
Urn Models ◽  
Pólya Urn ◽  
Polya Urn
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