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.

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.

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

scholarly journals Central limit theorems for generalized Pólya urn models

2006 ◽  
Vol 43 (4) ◽  
pp. 938-951 ◽  
Author(s):  
I. Higueras ◽  
J. Moler ◽  
F. Plo ◽  
M. San Miguel
Keyword(s):  
Covariance Matrix ◽  
Central Limit ◽  
Limit Theorems ◽  
Lyapunov Equation ◽  
Central Limit Theorems ◽  
Random Trees ◽  
Urn Models ◽  
Pólya Urn ◽  
Polya Urn ◽  
Generalized Pólya Urn

In this paper we obtain central limit theorems for generalized Pólya urn models with L ≥ 2 colors where one out of K different replacements (actions) is applied randomly at each step. Each possible action constitutes a row of the replacement matrix, which can be nonsquare and random. The actions are chosen following a probability distribution given by an arbitrary function of the proportions of the balls of different colors present in the urn. Moreover, under the same hypotheses it is proved that the covariance matrix of the asymptotic distribution is the solution of a Lyapunov equation, and a procedure is given to obtain the covariance matrix in an explicit form. Some applications of these results to random trees and adaptive designs in clinical trials are also presented.

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

scholarly journals On Generalized Pólya Urn Models

2013 ◽  
Vol 50 (4) ◽  
pp. 1169-1186 ◽  
Author(s):  
May-Ru Chen ◽  
Markus Kuba
Keyword(s):  
Discrete Time ◽  
Higher Moments ◽  
Urn Models ◽  
Urn Model ◽  
Time Step ◽  
Model Case ◽  
Pólya Urn ◽  
Polya Urn ◽  
Generalized Pólya Urn

We study an urn model introduced in the paper of Chen and Wei (2005), where at each discrete time step m balls are drawn at random from the urn containing colors white and black. Balls are added to the urn according to the inspected colors, generalizing the well known Pólya-Eggenberger urn model, case m = 1. We provide exact expressions for the expectation and the variance of the number of white balls after n draws, and determine the structure of higher moments. Moreover, we discuss extensions to more than two colors. Furthermore, we introduce and discuss a new urn model where the sampling of the m balls is carried out in a step-by-step fashion, and also introduce a generalized Friedman's urn model.

scholarly journals Central limit theorems for generalized Pólya urn models

2006 ◽  
Vol 43 (04) ◽  
pp. 938-951
Author(s):  
I. Higueras ◽  
J. Moler ◽  
F. Plo ◽  
M. San Miguel
Keyword(s):  
Covariance Matrix ◽  
Central Limit ◽  
Limit Theorems ◽  
Lyapunov Equation ◽  
Central Limit Theorems ◽  
Random Trees ◽  
Urn Models ◽  
Pólya Urn ◽  
Polya Urn ◽  
Generalized Pólya Urn

In this paper we obtain central limit theorems for generalized Pólya urn models with L ≥ 2 colors where one out of K different replacements (actions) is applied randomly at each step. Each possible action constitutes a row of the replacement matrix, which can be nonsquare and random. The actions are chosen following a probability distribution given by an arbitrary function of the proportions of the balls of different colors present in the urn. Moreover, under the same hypotheses it is proved that the covariance matrix of the asymptotic distribution is the solution of a Lyapunov equation, and a procedure is given to obtain the covariance matrix in an explicit form. Some applications of these results to random trees and adaptive designs in clinical trials are also presented.

scholarly journals Generalized Pólya urn Designs with Null Balance

2007 ◽  
Vol 44 (03) ◽  
pp. 661-669 ◽  
Author(s):  
Alessandro Baldi Antognini ◽  
Simone Giannerini
Keyword(s):  
Clinical Trials ◽  
Markov Process ◽  
Asymptotic Normality ◽  
Asymptotic Variance ◽  
Almost Sure Convergence ◽  
Natural Representation ◽  
Pólya Urn ◽  
Polya Urn ◽  
Generalized Pólya Urn ◽  
Replacement Rule

In this paper we propose a class of sequential urn designs based on generalized Pólya urn (GPU) models for balancing the allocations of two treatments in sequential clinical trials. In particular, we consider a GPU model characterized by a 2 x 2 random addition matrix with null balance (i.e. null row sums) and replacement rule depending upon the urn composition. Under this scheme, the urn process has a Markovian structure and can be regarded as a random extension of the classical Ehrenfest model. We establish almost sure convergence and asymptotic normality for the frequency of treatment allocations and show that in some peculiar cases the asymptotic variance of the design admits a natural representation based on the set of orthogonal polynomials associated with the corresponding Markov process.

On the local limit theorem for non-uniformly ergodic Markov chains

1995 ◽  
Vol 32 (1) ◽  
pp. 52-62 ◽  
Author(s):  
Marc Séva
Keyword(s):  
Markov Chain ◽  
Limit Theorem ◽  
Local Limit Theorem ◽  
Local Limit ◽  
Exponential Moment ◽  
Countable State Space ◽  
Uniform Ergodicity ◽  
Countable State ◽  
Reflected Random Walk ◽  
Positive Half

Using an approach similar to that of Guivarc'h and Hardy (1988), we show that the local limit theorem holds for a Markov chain on a countable state space, with non-uniform ergodicity, when the recurrence is fast enough. We present a detailed study of a typical example, the reflected random walk on the positive half-line with negative mean and finite exponential moment. The results can be extended to some random walks on ℕ.

Sign in / Sign up

Export Citation Format

Share Document