scholarly journals The Rescaled Pólya Urn and the Wright–Fisher Process with Mutation

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2909
Author(s):  
Giacomo Aletti ◽  
Irene Crimaldi
Keyword(s):  
Empirical Evidence ◽  
Almost Sure Convergence ◽  
Suitable Choice ◽  
Model Parameters ◽  
Reinforcement Mechanism ◽  
Pólya Urn ◽  
Polya Urn ◽  
Deterministic Limit ◽  
Fisher Process

In recent papers the authors introduce, study and apply a variant of the Eggenberger–Pólya urn, called the “rescaled” Pólya urn, which, for a suitable choice of the model parameters, exhibits a reinforcement mechanism mainly based on the last observations, a random persistent fluctuation of the predictive mean and the almost sure convergence of the empirical mean to a deterministic limit. In this work, motivated by some empirical evidence, we show that the multidimensional Wright–Fisher diffusion with mutation can be obtained as a suitable limit of the predictive means associated to a family of rescaled Pólya urns.

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.

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.

scholarly journals Generalized Pólya urn Designs with Null Balance

2007 ◽  
Vol 44 (3) ◽  
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.

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

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.

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